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

<b>let </b><font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">E</font> is <a href="ordinal1.html#V1">epsilon-transitive</a>
 ;<br/>

<b>thus </b><a NAME="E2:7"/>
( <font color="Maroon">E</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K9">the_axiom_of_infinity</a>  implies  ex <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/> ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Olive">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">E</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K9">the_axiom_of_infinity</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E3:7_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> holds <br/> ex <font color="Olive">w</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> st <br/>( <font color="Olive">v</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> ) ) )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">u</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E5:7_1"/>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E6:7_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E7:7_1"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">Y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E10:7_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">v</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">w</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>thus </b><a NAME="E12:7_1"/>
( <font color="Maroon">Y</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>


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

<b>given </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E3:7"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) ) )
 ;<br/>

<a NAME="E4:7"/>
 ex <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> st <br/>( <font color="Olive">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> holds <br/> ex <font color="Olive">w</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> st <br/>( <font color="Olive">v</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Olive">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">u</font> = <font color="Maroon">X</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">u</font>
;<br/>

<b>thus </b><a NAME="E2:7_2"/>
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font>;<br/>

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

<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:7_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">v</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:7_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i>, , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">w</font> = <font color="Maroon">Z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">w</font>
;<br/>

<b>thus </b><a NAME="E8:7_2"/>
( <font color="Maroon">v</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">w</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:7"/>
<font color="Maroon">E</font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K9">the_axiom_of_infinity</a> 
 <b>by </b><i>, </i>;<br/>


</div>
