<?xml version="1.0"?>
<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/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green" title="E41">A2</font></i>: 
<span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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>
 ;<br/>

<b>given </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E4:54"/><i><font color="Green" title="E42">A3</font></i>: 
( <font color="Maroon" title="c3">x</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; not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<a NAME="E5:54"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="classes1.html#E4:54"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<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="E7:54"/><i><font color="Green" title="E43">A4</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>and </b><br/><a NAME="E8:54"/><i><font color="Green" title="E44">A5</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font></span>)</span> or  not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )
 <b>by </b><i><a class="ref" href="tarski.html#T7" title="TARSKI:th.7">TARSKI:7</a></i>;<br/>
<a NAME="E9:54"/>
(  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font> is <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a> &amp; <font color="Maroon" title="c4">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> )
 
<b>by </b><i><a class="txt" href="classes1.html#E1:54"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="classes1.html#E7:54"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="classes1.html#T27" target="_self" title="CLASSES1:th.27">Th27</a></i>;<br/>
<a NAME="E10:54"/><b>then </b><i><font color="Green" title="E45">A6</font></i>: 
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="classes1.html#K1" title="CLASSES1:func.1">Tarski-Class</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E11:54"/>
<font color="Maroon" title="c4">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ;<br/>

<a NAME="E2:54_1"/><b>then </b>
( not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c2">A</font></span>)</span> &amp; <font color="Maroon" title="c5">x</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> )
 
<b>by </b><i><a class="txt" href="classes1.html#E8:54"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="classes1.html#E10:54"><i><font color="Green" title="E45">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:54_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:54"/><b>then </b>
( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font></span>)</span> &amp; <font color="Maroon" title="c4">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> )
 
<b>by </b><i><a class="txt" href="classes1.html#E7:54"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="classes1.html#T36" target="_self" title="CLASSES1:th.36">Th36</a></i>;<br/>
<a NAME="E13:54"/><b>then </b>
( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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> &amp; not <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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#E7:54"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:54"/>
contradiction
 <b>by </b><i><a class="txt" href="classes1.html#E2:54"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div>
