<?xml version="1.0"?>
<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="E1:48"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> is   <a href="card_1.html#NM1">Cardinal</a> )
 ;<br/>

<a NAME="E2:48"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> holds <br/>( <font color="Olive">x</font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Olive">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><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>assume </b><a NAME="E1:48_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 ;<br/>

<b>consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:48_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="card_lar.html#D4" target="_self">Def4</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:48_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E7:48_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E8:48_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">y</font> is   <a href="card_1.html#NM1">Cardinal</a>
 <b>by </b><i><a class="ref" href="card_lar.html#D3" target="_self">Def3</a>, <a class="ref" href="card_lar.html#T2" target="_self">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="card_1.html#NM1">Cardinal</a> <b>by </b><i/>;<br/>
<a NAME="E10:48_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D5" target="_self">Def5</a>, </i>;<br/>
<b>hence </b><a NAME="E11:48_1"/>
<font color="Maroon">x</font> is   <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="E12:48_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D5" target="_self">Def5</a>, , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E13:48_1"/>
<font color="Maroon">y1</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E14:48_1"/>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="card_lar.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">UNX</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T31">ORDINAL1:31</a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green">E50</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">UNX</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">UNX</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T24">CARD_1:24</a></i>;<br/>
<a NAME="E5:48"/>
<font color="Maroon">UNX</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">UNX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><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="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UNX</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="card_lar.html#D5" target="_self">Def5</a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>assume </b><a NAME="E3:48_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">UNX</font>
 ;<br/>

<a NAME="E4:48_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">UNX</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/>
<a NAME="E5:48_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">UNX</font> <a href="hidden.html#R2">in</a> <font color="Maroon">UNX</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#D5" target="_self">Def5</a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:48_2"/><i><font color="Green">E66</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">UNX</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E8:48_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:48_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E11:48_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E12:48_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">y</font> is   <a href="card_1.html#NM1">Cardinal</a>
 <b>by </b><i><a class="ref" href="card_lar.html#D3" target="_self">Def3</a>, <a class="ref" href="card_lar.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as   <a href="card_1.html#NM1">Cardinal</a> <b>by </b><i><a class="ref" href="card_lar.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E14:48_2"/><i><font color="Green">E71</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">UNX</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="ref" href="card_lar.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E15:48_2"/>
<font color="Maroon">y1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">UNX</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E16:48_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">y1</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">UNX</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>
<a NAME="E17:48_2"/><b>then </b>
<font color="Maroon">y1</font> <a href="ordinal1.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">UNX</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E18:48_2"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">UNX</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">UNX</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E19:48_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:48"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> is   <a href="card_1.html#NM1">Cardinal</a>
 <b>by </b><i><a class="ref" href="card_lar.html#D4" target="_self">Def4</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
