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<b>let </b><font color="Maroon">V</font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="zf_refle.html#NM1">Subclass</a> of <font color="Maroon">V</font>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">X</font> is <a href="zf_fund1.html#V8" target="_self">closed_wrt_A1-A7</a>
 ;<br/>

<b>consider </b><font color="Maroon">o</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">o</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> ;<br/>
<a NAME="E4:23"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="classes2.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">V</font></span>)</span>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">V</font></span>)</span>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">y</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> ;<br/>
<a NAME="E5:23"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">X</font> is <a href="zf_fund1.html#V1" target="_self">closed_wrt_A1</a>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E62</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">V</font></span>)</span>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">V</font></span>)</span>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">y</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">V</font></span>)</span>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">V</font></span>)</span>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">y</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> ;<br/><a NAME="E3:23_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">V</font></span>)</span>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">V</font></span>)</span>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">y</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:23_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">q</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#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">V</font></span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> &amp; <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> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="classes2.html#K1">}</a></span> )
 ;<br/><a NAME="E6:23_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:23_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:23"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, , </i>;<br/>


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