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<b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E2:38_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i><a class="txt" href="ordinal1.html#E4:38_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>] );<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:38_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E5:38_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:38_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="ordinal1.html#E6:38_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="ordinal1.html#E10:38_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E7:38_1"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E10:38_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E8:38_1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E10:38_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">C</font> <a href="ordinal1.html#R1" target="_self">c=</a> <font color="Olive">B</font> ) )
 <b>by </b><i><a class="txt" href="ordinal1.html#E13:38_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="ordinal1.html#E1:38"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<a NAME="E11:38_1"/>
 ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>] )
 
<b>by </b><i><a class="txt" href="ordinal1.html#E10:38_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="ordinal1.html#E14:38_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:38_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">C</font>]
 ;<br/>

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

<b>assume </b><a NAME="E13:38_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">B</font>]
 ;<br/>

<b>assume </b><a NAME="E14:38_1"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">C</font> <a href="ordinal1.html#R1" target="_self">c=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E15:38_1"/><b>then </b>
<font color="Maroon">B</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E16:38_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:38_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E10:38_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="ordinal1.html#E14:38_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E18:38_1"/><b>then </b>
<font color="Maroon">C</font> <a href="ordinal1.html#R1" target="_self">c=</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E19:38_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E10:38_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="ordinal1.html#E16:38_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="ordinal1.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E20:38_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal1.html#E14:38_1"><i><font color="Green">E26</font></i></a>, </i>;<br/>


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