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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E36">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> )
 ;<br/>

<a NAME="E2:48"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> holds <br/>( <font color="Olive" title="b1">x</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> &amp; <font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">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" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <b>and </b><br/><a NAME="E4:48_1"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:48_1"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:48_1"/><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E7:48_1"/><i><font color="Green" title="E41">A6</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">y</font>
 <b>and </b><br/><a NAME="E8:48_1"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c4">y</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:48"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="card_lar.html#E4:48_1"><i><font color="Green" title="E39">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">y1</font> = <font color="Maroon" title="c4">y</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="txt" href="card_lar.html#E8:48_1"><i><font color="Green" title="E42">A7</font></i></a></i>;<br/>
<a NAME="E10:48_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">y1</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:48_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="card_lar.html#E7:48_1"><i><font color="Green" title="E41">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:48_1"/>
<font color="Maroon" title="c2">x</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>

<a NAME="E12:48_1"/><i><font color="Green" title="E43">A8</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">y1</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E3:48_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="card_lar.html#E7:48_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E13:48_1"/>
<font color="Maroon" title="c5">y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E6:48_1"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E14:48_1"/>
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E12:48_1"><i><font color="Green" title="E43">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c2">UNX</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T31" title="ORDINAL1:th.31">ORDINAL1:31</a></i>;<br/>
<a NAME="E4:48"/><i><font color="Green" title="E37">A9</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">UNX</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">UNX</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T24" title="CARD_1:th.24">CARD_1:24</a></i>;<br/>
<a NAME="E5:48"/>
<font color="Maroon" title="c2">UNX</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">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" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green" title="E38">A10</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">UNX</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c3">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_lar.html#E1:48_2"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>assume </b><a NAME="E3:48_2"/>
not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">UNX</font>
 ;<br/>

<a NAME="E4:48_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">UNX</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">x1</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/>
<a NAME="E5:48_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">UNX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">UNX</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E1:48_2"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:48_2"/><i><font color="Green" title="E39">A11</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">UNX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>and </b><br/><a NAME="E8:48_2"/><i><font color="Green" title="E40">A12</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:48_2"/><i><font color="Green" title="E41">A13</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E11:48_2"/><i><font color="Green" title="E42">A14</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">y</font>
 <b>and </b><br/><a NAME="E12:48_2"/><i><font color="Green" title="E43">A15</font></i>: 
<font color="Maroon" title="c6">y</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>
 <b>by </b><i><a class="txt" href="card_lar.html#E1:48"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="card_lar.html#E8:48_2"><i><font color="Green" title="E40">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c6">y</font> as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> <b>by </b><i><a class="txt" href="card_lar.html#E12:48_2"><i><font color="Green" title="E43">A15</font></i></a></i>;<br/>
<a NAME="E14:48_2"/><i><font color="Green" title="E44">A16</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">UNX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">y1</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E7:48_2"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="card_lar.html#E11:48_2"><i><font color="Green" title="E42">A14</font></i></a></i>;<br/>
<a NAME="E15:48_2"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c7">y1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">UNX</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E10:48_2"><i><font color="Green" title="E41">A13</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E16:48_2"/><b>then </b>
<font color="Maroon" title="c7">y1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">UNX</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<a NAME="E17:48_2"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">UNX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">UNX</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E14:48_2"><i><font color="Green" title="E44">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:48_2"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:48"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">X</font> is   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a>
 <b>by </b><i><a class="txt" href="card_lar.html#E4:48"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
