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<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/>

<a NAME="E1:42"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> 0
 
;<br/>
<b>then </b><i><font color="Green" title="E26">A1</font></i>: 1 = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>

;<br/>
<a NAME="E3:42"/>
1 is <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c2">IT</font> = 1; <i><font color="Red">:: according to </font></i><a class="ref" href="card_1.html#D1" target="_self" title="CARD_1:def.1">CARD_1:def 1</a><br/>

<b>thus </b><a NAME="E1:42_2"/>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">IT</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E2:42_2"/>
<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c2">IT</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 ;<br/>

<b>then 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="E4:42_2"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="card_1.html#E2:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="card_1.html#T48" target="_self" title="CARD_1:th.48">Th48</a></i>;<br/>
<a NAME="E5:42_2"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="card_1.html#E4:42_2"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">y</font> as   <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="E7:42_2"/>
<font color="Maroon" title="c5">y</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="card_1.html#E5:42_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E8:42_2"/><b>then </b>
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> )
 
<b>by </b><i><a class="txt" href="card_1.html#E4:42_2"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></i>;<br/>
<b>hence </b><a NAME="E9:42_2"/>
<font color="Maroon" title="c2">IT</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="card_1.html#E2:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="card_1.html#E4:42_2"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="card_1.html#E5:42_2"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c2">M</font> = 1 as   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a> ;<br/>
<a NAME="E5:42"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">M</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="card_1.html#E2:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="card_1.html#T48" target="_self" title="CARD_1:th.48">Th48</a></i>;<br/>
<b>hence </b><a NAME="E6:42"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="card_1.html#D5" target="_self" title="CARD_1:def.5">Def5</a></i>;<br/>


</div>
