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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="classes2.html#NM1" title="CLASSES2:NM.1">Universe</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <a href="zf_refle.html#NM1" title="ZF_REFLE:NM.1">Subclass</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  holds <br/> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a>  implies  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:27"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c2">X</font> is  <a href="zf_fund1.html#V8" title="ZF_FUND1:attr.8">closed_wrt_A1-A7</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font></span><br/>

<span class="kw">assume </span><a NAME="E2:27"/>
 not  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:27"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 <span class="kw">and </span><br/><a NAME="E5:27"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
 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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">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"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
 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>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E33">A3</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">L</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E8:27"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">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="c4">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">M</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="#E6:27"><span class="lab"><font color="Green" title="E34">A4</font></span></a>);<br/></span>
<a NAME="E9:27"/>
<font color="Maroon" title="c4">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="zf_fund1.html#T3" target="_self" title="ZF_FUND1:th.3">Th3</a></span>;<br/>
<a NAME="E10:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A6</font></span>: 
 <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="c4">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal3.html#T8" title="ORDINAL3:th.8">ORDINAL3:8</a></span>;<br/>
<a NAME="E11:27"/>
 not  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A5</font></span></a></span>;<br/>
<a NAME="E12:27"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">L</font> is  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="ordinal1.html#D11" title="ORDINAL1:def.11">ORDINAL1:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">M</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E14:27"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T29" title="ORDINAL1:th.29">ORDINAL1:29</a></span>;<br/>
<a NAME="E15:27"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c5">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="ordinal1.html#T6" title="ORDINAL1:th.6">ORDINAL1:6</a></span>;<br/>
<a NAME="E16:27"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<font color="Maroon" title="c4">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A5</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E40">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:27_1"/>
 not <font color="Maroon" title="c5">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c4">L</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E16:27"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E15:27"><span class="lab"><font color="Green" title="E38">A8</font></span></a></span>;<br/><a NAME="E3:27_1"/>
<font color="Maroon" title="c5">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:27"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/><a NAME="E4:27_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><span class="kw">hence </span><a NAME="E5:27_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="ordinal1.html#T9" title="ORDINAL1:th.9">ORDINAL1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E18:27"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="zf_fund1.html#T2" target="_self" title="ZF_FUND1:th.2">Th2</a></span>;<br/>
<a NAME="E19:27"/><span class="kw">then </span>
<font color="Maroon" title="c5">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="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E17:27"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="zf_fund1.html#T4" target="_self" title="ZF_FUND1:th.4">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:27"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E14:27"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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