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<span class="kw">let </span><font color="Maroon" title="c1">U</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"> (  not <font color="Maroon" title="c1">U</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  iff  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:108"/>
(  not <font color="Maroon" title="c1">U</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  implies  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font> )
  <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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font> implies  not <font color="Maroon" title="c1">U</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:108_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">U</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="card_3.html#D14" title="CARD_3:def.14">CARD_3:def 14</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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font></span><br/>

<a NAME="E2:108_1"/><span class="kw">then </span>
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal6.html#T4" target="_self" title="ORDINAL6:th.4">EXCH1</a></span>;<br/>
<a NAME="E3:108_1"/><span class="kw">then </span>
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K2" title="ORDINAL1:func.2">On</a> <font color="Maroon" title="c1">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes2.html#T9" title="CLASSES2:th.9">CLASSES2:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:108_1"/>
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 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>

<span class="kw">assume </span><a NAME="E2:108"/>
 <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">U</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c1">U</font> is  <a href="card_3.html#V4" title="CARD_3:attr.4">countable</a> </span><br/>

<a NAME="E3:108"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="classes2.html#T1" title="CLASSES2:th.1">CLASSES2:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:108"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">U</font> <a href="matroid0.html#NR1" title="MATROID0:NR.1">c/=</a>  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="card_3.html#D14" title="CARD_3:def.14">CARD_3:def 14</a> <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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