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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c2">X</font> is <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>
 ;<br/>

<b>assume </b><a NAME="E2:27"/>
not  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:27"/><i><font color="Green" title="E32">A2</font></i>: 
( <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; not <font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">K</font> = <font color="Maroon" title="c3">o</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_fund1.html#E4:27"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; not $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> );<br/>
<a NAME="E6:27"/>
( <font color="Maroon" title="c4">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; not <font color="Maroon" title="c4">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:27"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/>
<a NAME="E7:27"/><b>then </b><i><font color="Green" title="E33">A3</font></i>: 
 ex <font color="Olive" title="b1">K</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">K</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c5">L</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E9:27"/><i><font color="Green" title="E34">A4</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">L</font>] &amp; ( for <font color="Olive" title="b1">M</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">M</font>] holds <br/><font color="Maroon" title="c5">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">M</font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="zf_fund1.html#E7:27"><i><font color="Green" title="E33">A3</font></i></a>);<br/></i>
<a NAME="E10:27"/>
<font color="Maroon" title="c5">L</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> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:27"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="zf_fund1.html#T3" target="_self" title="ZF_FUND1:th.3">Th3</a></i>;<br/>
<a NAME="E11:27"/><b>then </b><i><font color="Green" title="E35">A5</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/>
<a NAME="E12:27"/><i><font color="Green" title="E36">A6</font></i>: 
<font color="Maroon" title="c5">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E13:27"/><i><font color="Green" title="E37">A7</font></i>: 
<font color="Maroon" title="c5">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green" title="E34">A4</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E38">A8</font></i>: 
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">L</font>
 ;<br/><a NAME="E2:27_1"/>
<font color="Maroon" title="c5">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:27_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E13:27"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="zf_fund1.html#E1:27_1"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:27"/><b>then </b>
not <font color="Maroon" title="c5">L</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E11:27"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">M</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E17:27"/><i><font color="Green" title="E38">A9</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/>
<a NAME="E18:27"/><i><font color="Green" title="E39">A10</font></i>: 
<font color="Maroon" title="c6">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">L</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E17:27"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/>
<div><i><font color="Green" title="E40">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2"/>
not <font color="Maroon" title="c6">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/><a NAME="E2:27_2"/><b>then </b><i><font color="Green" title="E41">A12</font></i>: 
<font color="Maroon" title="c5">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">M</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="zf_fund1.html#E12:27"><i><font color="Green" title="E36">A6</font></i></a>, <a class="txt" href="zf_fund1.html#E18:27"><i><font color="Green" title="E39">A10</font></i></a></i>;<br/><a NAME="E3:27_2"/>
<font color="Maroon" title="c6">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">L</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E18:27"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/><a NAME="E4:27_2"/><b>then </b>
<font color="Maroon" title="c6">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:27_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E5:27_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E17:27"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="ordinal1.html#T14" title="ORDINAL1:th.14">ORDINAL1:14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:27"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">M</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:27"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></i>;<br/>
<a NAME="E21:27"/><b>then </b>
<font color="Maroon" title="c6">M</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">M</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:27"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="zf_fund1.html#E19:27"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="zf_fund1.html#T4" target="_self" title="ZF_FUND1:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E22:27"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="zf_fund1.html#E17:27"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></i>;<br/>


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