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<b>let </b><font color="Maroon" title="c1">x</font> be  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  <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><a NAME="E1:20"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E2:20"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c1">x</font>;<br/>
<a NAME="E4:20"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c1">x</font> <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="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c1">x</font> <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="E2:20_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal1.html#E1:20"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="xboole_1.html#T60" title="XBOOLE_1:th.60">XBOOLE_1:60</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:20"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c1">x</font>
 
;<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="E7:20"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c1">x</font>
 <b>and </b><br/><a NAME="E8:20"/><i><font color="Green" title="E15">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c1">x</font> or  not <font color="Olive" title="b1">a</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="E9:20"/><i><font color="Green" title="E16">A5</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font> )
 
<b>by </b><i><a class="txt" href="ordinal1.html#E7:20"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E10:20"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" target="_self" title="ORDINAL1:def.2">Def2</a>, <a class="txt" href="ordinal1.html#E7:20"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_2"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">y</font>
 ;<br/><a NAME="E2:20_2"/><b>then </b>
not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E8:20"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E3:20_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E10:20"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="ordinal1.html#E1:20_2"><i><font color="Green" title="E18">A7</font></i></a>, <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="E12:20"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:20_3"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:20_3"/><i><font color="Green" title="E19">A9</font></i>: 
( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">x</font> )
 ;<br/><a NAME="E4:20_3"/>
<font color="Maroon" title="c6">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E3:20_3"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="ordinal1.html#D2" target="_self" title="ORDINAL1:def.2">Def2</a></i>;<br/><a NAME="E5:20_3"/><b>then </b>
not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E7:20"><i><font color="Green" title="E14">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>hence </b><a NAME="E6:20_3"/>
<font color="Maroon" title="c5">a</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="txt" href="ordinal1.html#E2:20"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="ordinal1.html#E9:20"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="ordinal1.html#E3:20_3"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="ordinal1.html#D3" target="_self" title="ORDINAL1:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:20"/><b>then </b>
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E15:20"/>
<font color="Maroon" title="c1">x</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="txt" href="ordinal1.html#E9:20"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="ordinal1.html#E12:20"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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