<?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="E1:82"/><i><font color="Green">E25</font></i>: 
(  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" target="_self">the_rank_of</a> <span class="p2">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span></span>)</span> &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Olive">A</font> holds <br/> <a href="classes1.html#K6" target="_self">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Olive">A</font> ) )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:82"/>
 <a href="classes1.html#K6" target="_self">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_1"/>
 <a href="classes1.html#K6" target="_self">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:82_1"/><b>then </b>
 <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="txt" href="classes1.html#E5:82"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:82_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:82"/>
not  <a href="classes1.html#K6" target="_self">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<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="E5:82"/><i><font color="Green">E26</font></i>: 
 <a href="classes1.html#K6" target="_self">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</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="E7:82"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> &amp;  <a href="classes1.html#K6" target="_self">the_rank_of</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="classes1.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E8:82"/><i><font color="Green">E31</font></i>: 
(  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K1" target="_self">Tarski-Class</a> <font color="Maroon">X</font> &amp;  <a href="classes1.html#K6" target="_self">the_rank_of</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="classes1.html#T6" target="_self">Th6</a>, , </i>;<br/>
<a NAME="E9:82"/><b>then </b>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4" target="_self">Rank</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="classes1.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E10:82"/><b>then </b>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="classes1.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E11:82"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<b>hence </b><a NAME="E12:82"/>
contradiction
 ;<br/>


</div>
