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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be  non  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">M</font> is  <a href="card_lar.html#V4" title="CARD_LAR:attr.4">Mahlo</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="card_1.html#V2" title="CARD_1:attr.2">limit_cardinal</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:44"/>
<font color="Maroon" title="c1">M</font> is  <a href="card_lar.html#V4" title="CARD_LAR:attr.4">Mahlo</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font> is  <a href="card_1.html#V2" title="CARD_1:attr.2">limit_cardinal</a> </span><br/>

<a NAME="E2:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N</font> is  <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a>  </span>}</span>  <a href="card_lar.html#R4" title="CARD_LAR:pred.4">is_stationary_in</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_lar.html#D10" target="_self" title="CARD_LAR:def.10">Def10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">REG</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">N</font> where <font color="Olive" title="b1">N</font> is   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> : <font color="Olive" title="b1">N</font> is  <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a>  </span>}</span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="card_lar.html#D8" target="_self" title="CARD_LAR:def.8">Def8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:44"/>
not <font color="Maroon" title="c1">M</font> is  <a href="card_1.html#V2" title="CARD_1:attr.2">limit_cardinal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">M1</font> being   <a href="card_1.html#NM1" title="CARD_1:NM.1">Cardinal</a><span class="kw"> such that </span><br/><a NAME="E6:44"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
 <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c3">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E7:44"/>
<font color="Maroon" title="c3">M1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="card_1.html#T32" title="CARD_1:th.32">CARD_1:32</a></span>;<br/>
<a NAME="E8:44"/><span class="kw">then </span>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">M1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></span>;<br/>
<a NAME="E9:44"/><span class="kw">then </span>
( <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">M1</font></span>)</span> is  <a href="card_lar.html#V2" title="CARD_LAR:attr.2">closed</a>  &amp; <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">M1</font></span>)</span> is  <a href="card_lar.html#V1" title="CARD_LAR:attr.1">unbounded</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_lar.html#T13" target="_self" title="CARD_LAR:th.13">Th13</a></span>;<br/>
<a NAME="E10:44"/><span class="kw">then </span>
<font color="Maroon" title="c2">REG</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">M1</font></span>)</span></span>)</span> <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="#E2:44"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="card_lar.html#D8" target="_self" title="CARD_LAR:def.8">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">M2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E12:44"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c4">M2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">REG</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">M1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E13:44"/>
<font color="Maroon" title="c4">M2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">REG</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:44"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">N</font> being   <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>   <a href="card_1.html#V1" title="CARD_1:attr.1">cardinal</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E15:44"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c5">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">M2</font>
 <span class="kw">and </span><br/><a NAME="E16:44"/>
<font color="Maroon" title="c5">N</font> is  <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> 
 ;<br/>
<a NAME="E17:44"/>
<font color="Maroon" title="c4">M2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c3">M1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:44"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E18:44"/><span class="kw">then </span>
not <font color="Maroon" title="c5">N</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">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:44"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E19:44"/><span class="kw">then </span>
not <font color="Maroon" title="c5">N</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></span>;<br/>
<a NAME="E20:44"/><span class="kw">then </span>
<font color="Maroon" title="c3">M1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></span>;<br/>
<a NAME="E21:44"/><span class="kw">then </span>
( <font color="Maroon" title="c5">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="card_1.html#K2" title="CARD_1:func.2">nextcard</a> <font color="Maroon" title="c3">M1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">N</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T107" title="CARD_3:th.107">CARD_3:107</a></span>;<br/>
<a NAME="E22:44"/><span class="kw">then </span>
<font color="Maroon" title="c5">N</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E35">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:44"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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