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<b>let </b><font color="Maroon" title="c1">W</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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="c1">W</font> implies  ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b1">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b2">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="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">M</font> is <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a> ) )</span><br/>

<b>assume </b><a NAME="E1:38"/><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="c1">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font> ex <font color="Olive" title="b2">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>  st <br/>( <font color="Olive" title="b1">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b2">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="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">M</font> is <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a> )</span><br/>

<b>then consider </b><font color="Maroon" title="c2">a</font> being    <a href="ordinal4.html#M1" title="ORDINAL4:mode.1">Ordinal</a> of <font color="Maroon" title="c1">W</font>, <font color="Maroon" title="c3">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="E3:38"/><i><font color="Green" title="E28">A2</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">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="c2">a</font> &amp; <font color="Maroon" title="c3">M</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c1">W</font> )
 <b>by </b><i><a class="ref" href="zfrefle1.html#T41" target="_self" title="ZFREFLE1:th.41">Th41</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">a</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Olive" title="b1">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="c2">a</font> &amp; <font color="Olive" title="b1">M</font> is <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a> )</span><br/>

<b>take </b>
<font color="Maroon" title="c3">M</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">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="c2">a</font> &amp; <font color="Maroon" title="c3">M</font> is <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a> )</span><br/>

<a NAME="E4:38"/>
( <font color="Maroon" title="c3">M</font> is <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a> &amp; <font color="Maroon" title="c1">W</font> is <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a> &amp; <font color="Maroon" title="c1">W</font> <a href="zfrefle1.html#R2" title="ZFREFLE1:pred.2">&lt;==&gt;</a> <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="txt" href="#E1:38"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="#E3:38"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="zf_refle.html#T7" title="ZF_REFLE:th.7">ZF_REFLE:7</a></i>;<br/>
<b>hence </b><a NAME="E5:38"/>
( <font color="Maroon" title="c2">a</font> <a href="ordinal2.html#R2" title="ORDINAL2:pred.2">is_cofinal_with</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon" title="c3">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="c2">a</font> &amp; <font color="Maroon" title="c3">M</font> is <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a> )
 <b>by </b><i><a class="txt" href="#E3:38"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="zfrefle1.html#T10" target="_self" title="ZFREFLE1:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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