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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>;<br/>

<b>assume </b><a NAME="E1:3"/>
 <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>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">u</font> =  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c3">u</font> = <font color="Maroon" title="c2">u</font>;<br/><b>thus </b><a NAME="E1:3_1"/>
<font color="Maroon" title="c3">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> 
 ;<br/><b>let </b><font color="Maroon" title="c4">v</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font>;<br/><b>assume </b><a NAME="E2:3_1"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">v</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/><a NAME="E4:3_1"/><i><font color="Green" title="E5">A2</font></i>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></i>;<br/><a NAME="E5:3_1"/><b>then </b>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:3_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/><a NAME="E6:3_1"/><b>then </b>
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">u</font> &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> )
 <b>by </b><i><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="c6">w</font> =  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">W</font> <b>by </b><i><a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></i>;<br/><b>take </b><font color="Maroon" title="c7">w</font> = <font color="Maroon" title="c6">w</font>;<br/><a NAME="E8:3_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/><a NAME="E9:3_1"/><b>then </b>
( <font color="Maroon" title="c4">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">w</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">w</font> &amp; <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> )
 <b>by </b><i><a class="txt" href="zf_refle.html#E2:3_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="zf_refle.html#E4:3_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><b>hence </b><a NAME="E10:3_1"/>
( <font color="Maroon" title="c4">v</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c7">w</font> &amp; <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">u</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:3"/>
<font color="Maroon" title="c1">W</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="ref" href="zfmodel1.html#T6" title="ZFMODEL1:th.6">ZFMODEL1:6</a></i>;<br/>


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