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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E2:44"/><i><font color="Green">E26</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T38" target="_self">Th38</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:44"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E5:44"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E4:44"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E6:44"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:44"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E8:44"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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#E2:44"><i><font color="Green">E26</font></i></a>, <a class="txt" href="ordinal1.html#E4:44"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E10:44"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1" target="_self">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="ordinal1.html#E6:44"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal1.html#T32" target="_self">Th32</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E11:44"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E4:44"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ordinal1.html#E10:44"><i><font color="Green">E29</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E12:44"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>assume </b><a NAME="E13:44"/><i><font color="Green">E31</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E14:44"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E15:44"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E13:44"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal1.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E16:44"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E4:44"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ordinal1.html#E10:44"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E17:44"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1" target="_self">c=</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E18:44"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E4:44"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ordinal1.html#E12:44"><i><font color="Green">E30</font></i></a>, <a class="txt" href="ordinal1.html#E15:44"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:44"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal1.html#E10:44"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal1.html#E13:44"><i><font color="Green">E31</font></i></a></i>;<br/>


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