<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">W</font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">L</font> be   <a href="zf_refle.html#NM2">DOMAIN-Sequence</a> of <font color="Maroon">W</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:5"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">b</font> holds <br/><font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b</font>
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">a</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font> holds <br/> ( <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font> &amp; <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Olive">a</font> is <a href="ordinal1.html#V1">epsilon-transitive</a> )
 <b>and </b><br/><a NAME="E3:5"/><i><font color="Green">E49</font></i>: 
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font> is <a href="zf_fund2.html#V1" target="_self">predicatively_closed</a>
 ;<br/>

<a NAME="E4:5"/><i><font color="Green">E50</font></i>: 
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a><br/>

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:5_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:5_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="zf_refle.html#D5">ZF_REFLE:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<a NAME="E7:5_1"/>
<font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">u</font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:5_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">u</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>let </b><font color="Maroon">u1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E9:5_1"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E10:5_1"/><b>then </b>
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">u</font> &amp; <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">u</font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, <a class="ref" href="zf_refle.html#T24">ZF_REFLE:24</a></i>;<br/>
<b>hence </b><a NAME="E11:5_1"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">M</font> =  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font>;<br/>
<a NAME="E5:5"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">a</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font>  st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Olive">a</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Olive">a</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Olive">X</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font>;<br/>



<b>assume </b><a NAME="E1:5_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">X</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font> ;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font>;<br/>
<a NAME="E4:5_2"/>
<font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:5_2"/><b>then </b>
(  <a href="zf_fund2.html#K1" target="_self">Section</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="zf_lang.html#K11">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span> <a href="zf_lang.html#K5">'in'</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="zf_lang1.html#K1">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<font color="Maroon">e</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font> &amp; <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font> is <a href="ordinal1.html#V1">epsilon-transitive</a> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:5_2"/>
<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:5_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E2:5_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/><a NAME="E3:5_3"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">D</font> = <span class="p1">(<span class="default"><a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/><a NAME="E5:5_3"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  ex <font color="Olive">a</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">d</font>,<font color="Olive">a</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>

<a NAME="E1:5_3_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">u2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:5_3_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:5_3_1"/>
<font color="Maroon">u2</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="zf_refle.html#D5">ZF_REFLE:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u2</font> = <font color="Maroon">u2</font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<b>take </b>
<font color="Maroon">u2</font>
;<br/>

<b>thus </b><a NAME="E6:5_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">d</font>,<font color="Maroon">u2</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:5_3"/><i><font color="Green">E72</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  ex <font color="Olive">a</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">d</font>,<font color="Olive">a</font>] &amp; ( for <font color="Olive">b</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">d</font>,<font color="Olive">b</font>] holds <br/><font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">b</font> ) ) ) )
 <b>from </b><i><a class="ref" href="zf_refle.html#S2">ZF_REFLE:sch 2</a>();<br/></i><a NAME="E8:5_3"/><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:5_3_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:5_3_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u1</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<a NAME="E5:5_3_2"/>
 ex <font color="Olive">a</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u1</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Olive">a</font> &amp; ( for <font color="Olive">b</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font>  st <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">b</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:5_3_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="ordinal4.html#D2">ORDINAL4:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:5_3"/>
(  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <span class="p1">(<span class="default"><a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="classes2.html#T65">CLASSES2:65</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E10:5_3"/><b>then </b>
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="classes1.html#D1">CLASSES1:def 1</a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/><a NAME="E11:5_3"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">D</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_2.html#T3">CARD_2:3</a>, <a class="ref" href="classes2.html#T1">CLASSES2:1</a></i>;<br/><a NAME="E12:5_3"/><b>then </b>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a>  )
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/><a NAME="E13:5_3"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="classes1.html#T2">CLASSES1:2</a></i>;<br/><a NAME="E14:5_3"/><b>then </b>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="zf_refle.html#T28">ZF_REFLE:28</a></i>;<br/><b>then reconsider </b><font color="Maroon">a</font> =  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="ordinal4.html#D2">ORDINAL4:def 2</a></i>;<br/><a NAME="E16:5_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:5_3_3"/>
<font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">u2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E4:5_3_3"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c</font> &amp; ( for <font color="Olive">b</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font>  st <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Olive">b</font> holds <br/><font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Olive">b</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E5:5_3_3"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:5_3_3"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E7:5_3_3"/><b>then </b>
<font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">c</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:5_3_3"/>
<font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E17:5_3"/><b>then </b>
( <font color="Maroon">D</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> &amp; <font color="Maroon">D</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a>, <a class="ref" href="zf_refle.html#T24">ZF_REFLE:24</a></i>;<br/><a NAME="E18:5_3"/><b>then </b>
( <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E19:5_3"/><b>then </b>
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="zf_refle.html#K5">.</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E20:5_3"/>
<span class="p1">(<span class="default"><a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:5"/>
 <a href="zf_refle.html#K4">Union</a> <font color="Maroon">L</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K10">the_axiom_of_power_sets</a> 
 <b>by </b><i>, <a class="ref" href="zfmodel1.html#T9">ZFMODEL1:9</a></i>;<br/>


</div>
