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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:42"/>
1 <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> 0
 
;<br/>
<b>then </b><i><font color="Green">E33</font></i>: 1 = 
<a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>

;<br/>
<a NAME="E3:42"/>
1 is <a href="card_1.html#V1" target="_self">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">IT</font> = 1; <i><font color="Red">:: according to </font></i><a class="ref" href="card_1.html#D1" target="_self">CARD_1:def 1</a><br/>

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

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

<b>assume </b><a NAME="E2:42_2"/>
<font color="Maroon">A</font>,<font color="Maroon">IT</font> <a href="wellord2.html#R2">are_equipotent</a> 
 ;<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="E4:42_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:42_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</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="E7:42_2"/>
<font color="Maroon">y</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E8:42_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<b>hence </b><a NAME="E9:42_2"/>
<font color="Maroon">IT</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = 1 as   <a href="card_1.html#NM1">Cardinal</a> ;<br/>
<a NAME="E5:42"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">M</font> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:42"/>
 <a href="card_1.html#K1" target="_self">Card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/>


</div>
