<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>thus </b><a NAME="E1:6"/>
<span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="relset_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E4:6_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="relset_2.html#E3:6_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:6_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 
<b>by </b><i><a class="txt" href="relset_2.html#E3:6_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="relset_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E6:6_1"/>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="relset_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>thus </b><a NAME="E4:6_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="relset_2.html#E4:6_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="relset_2.html#E3:6_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="relset_2.html#E3:6_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:6_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 <b>by </b><i><a class="txt" href="relset_2.html#E5:6_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:6"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 ;<br/>

<a NAME="E3:6"/><b>then </b><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="relset_2.html#NK1" target="_self">}_}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:6"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <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="ref" href="relset_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E6:6"/>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="relset_2.html#E3:6"><i><font color="Green">E5</font></i></a>, <a class="txt" href="relset_2.html#E5:6"><i><font color="Green">E6</font></i></a>, <a class="ref" href="relset_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:6"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="relset_2.html#E5:6"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:6"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="relset_2.html#NK1" target="_self">{_{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="relset_2.html#NK1" target="_self">}_}</a></span>
 <b>by </b><i><a class="txt" href="relset_2.html#E5:6"><i><font color="Green">E6</font></i></a>, <a class="ref" href="relset_2.html#T1" target="_self">Th1</a></i>;<br/>


</div>
