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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:74"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E2:74"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:74_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<a NAME="E2:74_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<b>assume </b><a NAME="E3:74_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E4:74_1"/><b>then </b>
(  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E81"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<a NAME="E5:74_1"/><b>then </b><i><font color="Green">E56</font></i>: 
(  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
<b>by </b><i><a class="txt" href="classes2.html#E2:74_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="classes1.html#T5">CLASSES1:5</a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<a NAME="E6:74_1"/><b>then </b>
(  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E1:74_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:74_1"/>
 <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="classes2.html#E5:74_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="classes2.html#T59" target="_self">Th59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:74"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:74_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:74_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/> <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E3:74_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E4:74_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E1:74_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E6:74_2"/><i><font color="Green">E57</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<i><font color="Green">E58</font></i>:  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="classes2.html#K15" target="_self">Universe_closure</a> <span class="p1">(<span class="default"><a href="card_3.html#K3">Union</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="classes2.html#E1:74_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="classes2.html#E6:74_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="classes2.html#E81"><i><font color="Green">Lemma49</font></i></a></i>
<br/>.= 
 <a href="classes2.html#K15" target="_self">Universe_closure</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>
;<br/>
<a NAME="E8:74_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="classes2.html#E4:74_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="classes2.html#E6:74_2"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E9:74_2"/><b>then </b>
 <a href="classes2.html#K16" target="_self">UNIVERSE</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="classes2.html#E4:74_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="classes2.html#E6:74_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:74_2"/><b>then </b>
(  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E7:74_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="classes2.html#D4" target="_self">Def4</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E11:74_2"/><b>then </b>
(  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K1">Tarski-Class</a> <span class="p1">(<span class="default"><a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E81"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="classes1.html#T5">CLASSES1:5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E12:74_2"/>
 <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:74"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="classes2.html#E1:74"><i><font color="Green">E52</font></i></a>, <a class="txt" href="classes2.html#E2:74"><i><font color="Green">E53</font></i></a>, <a class="txt" href="classes2.html#E3:74"><i><font color="Green">E54</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:74"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> implies  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E6:74"/><i><font color="Green">E56</font></i>: 
(  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes2.html#K16" target="_self">UNIVERSE</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E7:74"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>1</sub></font> iff ( <font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E8:74"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="classes2.html#E6:74"><i><font color="Green">E56</font></i></a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/>
<b>hence </b><a NAME="E9:74"/>
contradiction
 <b>by </b><i><a class="txt" href="classes2.html#E4:74"><i><font color="Green">E55</font></i></a>, <a class="txt" href="classes2.html#E6:74"><i><font color="Green">E56</font></i></a></i>;<br/>


</div>
