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<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E25">A1</font></i>: 
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0
 ;<br/>

<a NAME="E2:35"/>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0</span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0
 
<b>by </b><i><a class="ref" href="card_5.html#D2" target="_self" title="CARD_5:def.2">Def2</a></i>;<br/>
<a NAME="E3:35"/><b>then </b>
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0</span>)</span> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0
 
<b>by </b><i><a class="txt" href="card_5.html#E1:35"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c1">B</font> =  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0</span>)</span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="card_4.html#T9" title="CARD_4:th.9">CARD_4:9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c2">n</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">B</font>;<br/>
<a NAME="E5:35"/>
(  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p2">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p1">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0</span>)</span> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">B</font></span>)</span> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <span class="p2">(<span class="default"><a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">B</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<a NAME="E6:35"/><b>then </b>
 <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0 <a href="zfrefle1.html#R4" title="ZFREFLE1:pred.4">is_cofinal_with</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">B</font>
 
<b>by </b><i><a class="ref" href="card_5.html#D2" target="_self" title="CARD_5:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">xi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E8:35"/><i><font color="Green" title="E26">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c1">B</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0 &amp; <font color="Maroon" title="c3">xi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> &amp;  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c3">xi</font> )
 <b>by </b><i><a class="ref" href="zfrefle1.html#D5" title="ZFREFLE1:def.5">ZFREFLE1:def 5</a></i>;<br/>
<a NAME="E9:35"/><i><font color="Green" title="E27">A3</font></i>: 
(  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font></span>)</span> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <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="ref" href="card_1.html#T83" title="CARD_1:th.83">CARD_1:83</a>, <a class="ref" href="ordinal2.html#T26" title="ORDINAL2:th.26">ORDINAL2:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">rxi</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="card_5.html#E8:35"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="finset_1.html#T26" title="FINSET_1:th.26">FINSET_1:26</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1;<br/>
<a NAME="E11:35"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c4">rxi</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="card_5.html#E8:35"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="card_5.html#E9:35"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/>
<a NAME="E12:35"/><i><font color="Green" title="E29">A5</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E13:35"/><i><font color="Green" title="E30">A6</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>]
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:35"/><i><font color="Green" title="E31">A7</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">rxi</font> &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">rxi</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> holds <br/> not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">x</font>] ) )
 <b>from </b><i><a class="ref" href="card_3.html#S2" title="CARD_3:sch.2">CARD_3:sch 2</a>(<a class="txt" href="card_5.html#E11:35"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="card_5.html#E12:35"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="card_5.html#E13:35"><i><font color="Green" title="E30">A6</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="card_5.html#E8:35"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="card_5.html#E15:35"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font>
 ;<br/><a NAME="E2:35_1"/><b>then </b>
( <font color="Maroon" title="c7">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">x</font> or  not <font color="Maroon" title="c6">x</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )
 <b>by </b><i><a class="txt" href="card_5.html#E15:35"><i><font color="Green" title="E31">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E3:35_1"/>
<font color="Maroon" title="c7">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="c6">x</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:35"/><b>then </b>
(  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">x</font> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">x</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="card_5.html#E8:35"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="card_5.html#E9:35"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="card_5.html#E15:35"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="card_5.html#E40"><i><font color="Green" title="E23">Lm3</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a>, <a class="ref" href="ordinal2.html#T28" title="ORDINAL2:th.28">ORDINAL2:28</a></i>;<br/>
<b>hence </b><a NAME="E19:35"/>
contradiction
 <b>by </b><i><a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a></i>;<br/>


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