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<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">W</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E27">A1</font></i>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font>
 ;<br/>

<b>assume </b><a NAME="E2:39"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font>
 ;<br/>

<a NAME="E3:39"/><b>then </b><i><font color="Green" title="E28">A2</font></i>: 
(  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Maroon" title="c2">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c2">W</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="classes1.html#T76" title="CLASSES1:th.76">CLASSES1:76</a>, <a class="ref" href="classes2.html#T54" title="CLASSES2:th.54">CLASSES2:54</a></i>;<br/>
<a NAME="E4:39"/><b>then </b>
 <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="ref" href="classes1.html#T72" title="CLASSES1:th.72">CLASSES1:72</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">a</font> =  <a href="classes1.html#K6" title="CLASSES1:func.6">the_rank_of</a> <font color="Maroon" title="c1">X</font> as    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c2">W</font> <b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">b</font> being    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c2">W</font>, <font color="Maroon" title="c5">M</font> being  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> <b> such that </b><br/><a NAME="E7:39"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c5">M</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c2">W</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1:39"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="zfrefle1.html#T40" target="_self" title="ZFREFLE1:th.40">Th40</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">M</font>
;<br/>

<a NAME="E8:39"/>
( <font color="Maroon" title="c1">X</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="c3">a</font> &amp;  <a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c3">a</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E7:39"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="classes1.html#D8" title="CLASSES1:def.8">CLASSES1:def 8</a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<a NAME="E9:39"/><b>then </b>
( <font color="Maroon" title="c1">X</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="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c3">a</font></span>)</span> &amp;  <a href="classes1.html#K4" title="CLASSES1:func.4">Rank</a> <span class="p1">(<span class="default"><a href="ordinal4.html#K6" title="ORDINAL4:func.6">succ</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">M</font> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E7:39"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="classes1.html#T36" title="CLASSES1:th.36">CLASSES1:36</a>, <a class="ref" href="classes1.html#T43" title="CLASSES1:th.43">CLASSES1:43</a></i>;<br/>
<b>hence </b><a NAME="E10:39"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">M</font>
 ;<br/>

<a NAME="E11:39"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K3" title="ORDINAL1:func.3">On</a> <font color="Maroon" title="c2">W</font>
 
<b>by </b><i><a class="ref" href="zf_refle.html#T8" title="ZF_REFLE:th.8">ZF_REFLE:8</a></i>;<br/>
<b>hence </b><a NAME="E12:39"/>
<font color="Maroon" title="c5">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E3:39"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="zfrefle1.html#E7:39"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="classes1.html#T42" title="CLASSES1:th.42">CLASSES1:42</a></i>;<br/>

<a NAME="E13:39"/>
<font color="Maroon" title="c2">W</font> <a href="zf_model.html#NR3" title="ZF_MODEL:NR.3">is_a_model_of_ZF</a> 
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1:39"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="zf_refle.html#T7" title="ZF_REFLE:th.7">ZF_REFLE:7</a></i>;<br/>
<a NAME="E14:39"/><b>then </b>
<font color="Maroon" title="c2">W</font> <a href="zfrefle1.html#R1" title="ZFREFLE1:pred.1">|=</a>  <a href="zfrefle1.html#K2" title="ZFREFLE1:func.2">ZF-axioms</a> 
 
<b>by </b><i><a class="ref" href="zfrefle1.html#T4" target="_self" title="ZFREFLE1:th.4">Th4</a></i>;<br/>
<a NAME="E15:39"/><b>then </b>
( <font color="Maroon" title="c5">M</font> <a href="zfrefle1.html#R1" title="ZFREFLE1:pred.1">|=</a>  <a href="zfrefle1.html#K2" title="ZFREFLE1:func.2">ZF-axioms</a>  &amp; <font color="Maroon" title="c5">M</font> is <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E7:39"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="zfrefle1.html#T8" target="_self" title="ZFREFLE1:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E16:39"/>
<font color="Maroon" title="c5">M</font> <a href="zf_model.html#NR3" title="ZF_MODEL:NR.3">is_a_model_of_ZF</a> 
 <b>by </b><i><a class="ref" href="zfrefle1.html#T5" target="_self" title="ZFREFLE1:th.5">Th5</a></i>;<br/>


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