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<b>let </b><font color="Maroon">A</font> be  <a href="ordinal1.html#V4">being_limit_ordinal</a> <a href="finset_1.html#V1">infinite</a> <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/>



<b>thus </b><a NAME="E1:14"/>
( <font color="Maroon">X</font> is <a href="card_lar.html#V1" target="_self">unbounded</a> implies for <font color="Olive">B1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">C</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">X</font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 ;<br/>

<b>let </b><font color="Maroon">B1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E2:14_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:14_1"/>
not <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:14_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E6:14_1"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>thus </b><a NAME="E8:14_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E9:14_1"/>
<font color="Maroon">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:14"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">B1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">C</font> )
 ;<br/>

<b>assume </b><a NAME="E3:14"/>
not <font color="Maroon">X</font> is <a href="card_lar.html#V1" target="_self">unbounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">B1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:14"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E6:14"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:14"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E9:14"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">B1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E10:14"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:14"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:14"/>
contradiction
 ;<br/>


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