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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">B3</font>, <font color="Maroon" title="c3">B1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c4">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>;<br/>







<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c2">B3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B1</font>
 ;<br/>

<b>assume </b><a NAME="E2:29"/>
not <font color="Maroon" title="c2">B3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">B1</font>
 ;<br/>

<b>then 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="E4:29"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B3</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="card_lar.html#K2" title="CARD_LAR:func.2">limpoints</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <b>and </b><br/><a NAME="E5:29"/><i><font color="Green" title="E25">A3</font></i>: 
not <font color="Maroon" title="c5">x</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="c3">B1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:29"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B3</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:29"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">B3</font> <b>by </b><i><a class="txt" href="card_lar.html#E4:29"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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> ] <b>means </b>( not $1 is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; $1 is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> &amp;  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E8:29"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B2</font> where <font color="Olive" title="b1">B2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B2</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_lar.html#E4:29"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:29"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E5:29"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/>
<a NAME="E10:29"/><b>then </b><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<a NAME="E11:29"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x1</font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1" title="CARD_FIL:sch.1">CARD_FIL:sch 1</a>(<a class="txt" href="card_lar.html#E8:29"><i><font color="Green" title="E27">A5</font></i></a>);<br/></i>
<b>then reconsider </b><font color="Maroon" title="c7">x2</font> = <font color="Maroon" title="c6">x1</font> as  <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">being_limit_ordinal</a> <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">x2</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c7">x2</font> <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E14:29"/>
<font color="Maroon" title="c8">Y</font> is <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>
 
<b>by </b><i><a class="txt" href="card_lar.html#E11:29"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="card_lar.html#D4" target="_self" title="CARD_LAR:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E16:29"/><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y</font>
 <b>and </b><br/><a NAME="E17:29"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="txt" href="card_lar.html#E10:29"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="card_lar.html#T7" target="_self" title="CARD_LAR:th.7">Th7</a></i>;<br/>
<a NAME="E18:29"/><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E16:29"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:29"/>
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B3</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E6:29"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="card_lar.html#E16:29"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E20:29"/><b>then </b>
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c2">B3</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E18:29"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E21:29"/><b>then </b>
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B1</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E1:29"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<a NAME="E22:29"/><b>then </b>
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 
<b>by </b><i><a class="txt" href="card_lar.html#E17:29"><i><font color="Green" title="E31">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E23:29"/>
contradiction
 ;<br/>


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