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

<b>let </b><font color="Maroon" title="c6">E</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c6">E</font> is <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>
 ;<br/>

<b>thus </b><a NAME="E2:7"/>
( <font color="Maroon" title="c6">E</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K9" title="ZF_MODEL:func.9">the_axiom_of_infinity</a>  implies  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> &amp; <font color="Olive" title="b1">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>  &amp; ( for <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/> ex <font color="Olive" title="b3">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">Y</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b3">Z</font> &amp; <font color="Olive" title="b3">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon" title="c6">E</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K9" title="ZF_MODEL:func.9">the_axiom_of_infinity</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font><b> such that </b><br/><a NAME="E3:7_1"/><i><font color="Green" title="E5">A2</font></i>: 
( <font color="Maroon" title="c7">u</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>  &amp; ( for <font color="Olive" title="b1">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> holds <br/> ex <font color="Olive" title="b2">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> st <br/>( <font color="Olive" title="b1">v</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">w</font> &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> ) ) )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="zfmodel1.html#T6" target="_self" title="ZFMODEL1:th.6">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">X</font> = <font color="Maroon" title="c7">u</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c8">X</font>
;<br/>

<b>thus </b><a NAME="E5:7_1"/>
( <font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> &amp; <font color="Maroon" title="c8">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>  )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:7_1"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c9">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E6:7_1"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 ;<br/>

<a NAME="E7:7_1"/>
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c9">Y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> <b>by </b><i><a class="txt" href="zfmodel1.html#E6:7_1"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font><b> such that </b><br/><a NAME="E10:7_1"/><i><font color="Green" title="E7">A4</font></i>: 
( <font color="Maroon" title="c10">v</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c11">w</font> &amp; <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E3:7_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="zfmodel1.html#E6:7_1"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">w</font> = <font color="Maroon" title="c11">w</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c12">w</font>
;<br/>

<b>thus </b><a NAME="E12:7_1"/>
( <font color="Maroon" title="c9">Y</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c12">w</font> &amp; <font color="Maroon" title="c12">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E10:7_1"><i><font color="Green" title="E7">A4</font></i></a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c7">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E4:7"/><i><font color="Green" title="E5">A5</font></i>: 
( <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">E</font> &amp; <font color="Maroon" title="c7">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>  &amp; ( for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> holds <br/> ex <font color="Olive" title="b2">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Y</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">Z</font> &amp; <font color="Olive" title="b2">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> ) ) )
 ;<br/>

<a NAME="E5:7"/>
 ex <font color="Olive" title="b1">u</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> st <br/>( <font color="Olive" title="b1">u</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>  &amp; ( for <font color="Olive" title="b2">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> holds <br/> ex <font color="Olive" title="b3">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> st <br/>( <font color="Olive" title="b2">v</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b3">w</font> &amp; <font color="Olive" title="b3">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c8">u</font> = <font color="Maroon" title="c7">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> <b>by </b><i><a class="txt" href="zfmodel1.html#E4:7"><i><font color="Green" title="E5">A5</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">u</font>
;<br/>

<b>thus </b><a NAME="E2:7_2"/>
<font color="Maroon" title="c8">u</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> 
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:7"><i><font color="Green" title="E5">A5</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c9">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font>;<br/>

<b>assume </b><a NAME="E3:7_2"/>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">u</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:7_2"/><i><font color="Green" title="E6">A6</font></i>: 
( <font color="Maroon" title="c9">v</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c10">Z</font> &amp; <font color="Maroon" title="c10">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E4:7"><i><font color="Green" title="E5">A5</font></i></a></i>;<br/>
<a NAME="E6:7_2"/>
<font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">E</font>
 
<b>by </b><i><a class="txt" href="zfmodel1.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="zfmodel1.html#E4:7"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">w</font> = <font color="Maroon" title="c10">Z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">E</font> <b>by </b><i><a class="txt" href="zfmodel1.html#E5:7_2"><i><font color="Green" title="E6">A6</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">w</font>
;<br/>

<b>thus </b><a NAME="E8:7_2"/>
( <font color="Maroon" title="c9">v</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c11">w</font> &amp; <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">u</font> )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E5:7_2"><i><font color="Green" title="E6">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:7"/>
<font color="Maroon" title="c6">E</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K9" title="ZF_MODEL:func.9">the_axiom_of_infinity</a> 
 <b>by </b><i><a class="txt" href="zfmodel1.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="zfmodel1.html#T6" target="_self" title="ZFMODEL1:th.6">Th6</a></i>;<br/>


</div>
