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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">W</font> be   <a href="classes2.html#NM1">Universe</a>;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green">E37</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<b>assume </b><a NAME="E2:47"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E3:47"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">W</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3">On</a> <font color="Maroon">W</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="classes1.html#T76">CLASSES1:76</a>, <a class="ref" href="classes2.html#T54">CLASSES2:54</a></i>;<br/>
<a NAME="E4:47"/><b>then </b>
 <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="classes1.html#T72">CLASSES1:72</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> =  <a href="classes1.html#K6">the_rank_of</a> <font color="Maroon">X</font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">W</font>, <font color="Maroon">M</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:47"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">b</font> &amp; <font color="Maroon">M</font> <a href="zfrefle1.html#R2" target="_self">&lt;==&gt;</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="ref" href="zfrefle1.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<a NAME="E8:47"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K4">Rank</a> <font color="Maroon">a</font> &amp;  <a href="ordinal4.html#K6">succ</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="classes1.html#D8">CLASSES1:def 8</a>, <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E9:47"/><b>then </b>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">a</font></span>)</span> &amp;  <a href="classes1.html#K4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6">succ</a> <font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, <a class="ref" href="classes1.html#T36">CLASSES1:36</a>, <a class="ref" href="classes1.html#T43">CLASSES1:43</a></i>;<br/>
<b>hence </b><a NAME="E10:47"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E11:47"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8">ZF_REFLE:8</a></i>;<br/>
<b>hence </b><a NAME="E12:47"/>
<font color="Maroon">M</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="classes1.html#T42">CLASSES1:42</a></i>;<br/>

<a NAME="E13:47"/>
<font color="Maroon">W</font> <a href="zf_model.html#NR3" target="_self">is_a_model_of_ZF</a> 
 
<b>by </b><i><a class="ref" href="zfrefle1.html#D1" target="_self">Def1</a>, <a class="ref" href="zf_refle.html#T7">ZF_REFLE:7</a></i>;<br/>
<a NAME="E14:47"/><b>then </b>
<font color="Maroon">W</font> <a href="zfrefle1.html#R1" target="_self">|=</a>  <a href="zfrefle1.html#K2" target="_self">ZF-axioms</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E15:47"/><b>then </b>
( <font color="Maroon">M</font> <a href="zfrefle1.html#R1" target="_self">|=</a>  <a href="zfrefle1.html#K2" target="_self">ZF-axioms</a>  &amp; <font color="Maroon">M</font> is <a href="ordinal1.html#V1">epsilon-transitive</a> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E16:47"/>
<font color="Maroon">M</font> <a href="zf_model.html#NR3" target="_self">is_a_model_of_ZF</a> 
 <b>by </b><i/>;<br/>


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