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<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="E1:81"/><i><font color="Green" title="E59">A1</font></i>: 
(  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p2">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Olive" title="b1">A</font> holds <br/> <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) )
 
<b>by </b><i><a class="ref" href="classes1.html#D8" target="_self" title="CLASSES1:def.8">Def8</a></i>;<br/>
<b>thus </b><a NAME="E2:81"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1"/>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:81_1"/><b>then </b>
 <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="classes1.html#D8" target="_self" title="CLASSES1:def.8">Def8</a>, <a class="txt" href="classes1.html#E40"><i><font color="Green" title="E26">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:81_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>


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

<b>assume </b><a NAME="E3:81"/>
not  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:81"/><i><font color="Green" title="E60">A2</font></i>: 
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<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="E7:81"/><i><font color="Green" title="E61">A3</font></i>: 
( <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="classes1.html#E5:81"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="classes1.html#T80" target="_self" title="CLASSES1:th.80">Th80</a></i>;<br/>
<a NAME="E8:81"/><i><font color="Green" title="E62">A4</font></i>: 
(  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> )
 
<b>by </b><i><a class="txt" href="classes1.html#E7:81"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="classes1.html#T7" target="_self" title="CLASSES1:th.7">Th7</a>, <a class="ref" href="classes1.html#T71" target="_self" title="CLASSES1:th.71">Th71</a></i>;<br/>
<a NAME="E9:81"/><b>then </b>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="classes1.html#E1:81"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="classes1.html#E5:81"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="classes1.html#T36" target="_self" title="CLASSES1:th.36">Th36</a></i>;<br/>
<a NAME="E10:81"/><b>then </b>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="classes1.html#E8:81"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="classes1.html#D8" target="_self" title="CLASSES1:def.8">Def8</a></i>;<br/>
<a NAME="E11:81"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<b>hence </b><a NAME="E12:81"/>
contradiction
 ;<br/>


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