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

<b>assume </b><a NAME="E1:103"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 ;<br/>

<b>assume </b><a NAME="E2:103"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:103"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:103"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E30"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:103"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span>
 <b>and </b><br/><a NAME="E8:103"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span> or  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="tarski.html#T7">TARSKI:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:103_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:103_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T17" target="_self">Th17</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:103_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E5:103_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E39"><i><font color="Green">Lemma51</font></i></a></i>;<br/><a NAME="E6:103_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E7:103_1"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E36"><i><font color="Green">Lemma50</font></i></a>, <a class="txt" href="zfmisc_1.html#E2"><i><font color="Green">Lemma25</font></i></a></i>;<br/><a NAME="E8:103_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T26" target="_self">Th26</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E9:103_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E10:103_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E34"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="zfmisc_1.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:103"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E32"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:103"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E13:103"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E14:103"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T17" target="_self">Th17</a>, <a class="txt" href="zfmisc_1.html#E39"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:103"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E17:103"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E18:103"/>
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E45"><i><font color="Green">Lemma53</font></i></a></i>;<br/>
<a NAME="E19:103"/><b>then </b>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="txt" href="zfmisc_1.html#E36"><i><font color="Green">Lemma50</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E20:103"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T26" target="_self">Th26</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E21:103"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E34"><i><font color="Green">Lemma49</font></i></a></i>;<br/>


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