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

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

<b>assume </b><a NAME="E1:33"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">W</font> <a href="classes1.html#NR1" target="_self">is_Tarski-Class</a> 
 ;<br/>

<b>set </b><font color="Maroon">B</font> =  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span>;<br/>
<b>thus </b><a NAME="E2:33"/>
for <font color="Olive">X</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span> &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">X</font> holds <br/><font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T47">CLASSES1:47</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="classes1.html#D1">CLASSES1:def 1</a>,<a class="ref" href="classes1.html#D2">CLASSES1:def 2</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:33_1"/><b>then </b>
not <font color="Maroon">W</font>, <a href="xboole_0.html#K1">{}</a>  <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T46">CARD_1:46</a></i>;<br/><a NAME="E3:33_1"/><b>then </b><i><font color="Green">E28</font></i>: 
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>by </b><i><a class="txt" href="classes2.html#E3:33_1"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/><b>thus </b><a NAME="E4:33_1"/>
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="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span> holds <br/> <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_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="E1:33_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:33_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="classes2.html#E3:33_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="classes1.html#T35">CLASSES1:35</a></i>;<br/>
<a NAME="E4:33_1_1"/>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">W</font> &amp;  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="classes2.html#E3:33_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="classes2.html#T3" target="_self">Th3</a>, <a class="ref" href="classes1.html#T48">CLASSES1:48</a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1_1"/>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="classes2.html#E3:33_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="classes1.html#T35">CLASSES1:35</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:33"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span> or  <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span> or <font color="Olive">b<sub>1</sub></font>, <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span> <a href="tarski.html#R2">are_equipotent</a>  or <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">W</font></span>)</span> ) ) )
 <b>by </b><i>, <a class="ref" href="classes2.html#T1" target="_self">Th1</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="classes1.html#T33">CLASSES1:33</a></i>;<br/>


</div>
