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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<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">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font>,<font color="Olive">n</font> <a href="wellord2.html#R2">are_equipotent</a> ;<br/>
<b>assume </b><a NAME="E1:57"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:57"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i/>;<br/>
<a NAME="E3:57"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">Z</font>, <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">Z</font></span><a href="tarski.html#K1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:57_1"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> )
 ;<br/>

<b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:57_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Y</font>,<font color="Maroon">n</font> <a href="wellord2.html#R2">are_equipotent</a> 
 ;<br/>

<a NAME="E3:57_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Z</font></span><a href="tarski.html#K1">}</a></span>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:57_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Z</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E3:57_1_1"/>
<font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Z</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">n</font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T8" target="_self">Th8</a>, <a class="ref" href="card_1.html#T21" target="_self">Th21</a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:57_1"/>
( not <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Z</font></span><a href="tarski.html#K1">}</a></span>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1_2"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>take </b><font color="Maroon">m</font> = <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1;<br/>

<i><font color="Green">E54</font></i>: <font color="Maroon">m</font> = 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">n</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>
;<br/>
<a NAME="E3:57_1_2"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Z</font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span> <a href="wellord2.html#R2">are_equipotent</a> 
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1_2_2"/>
<font color="Maroon">n</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:57_1_2_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">n</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:57_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="card_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:57_1_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="card_1.html#T24" target="_self">Th24</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_3"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Z</font></span><a href="tarski.html#K1">}</a></span>;<br/><b>assume </b><a NAME="E2:57_1_2_3"/>
<font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Z</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:57_1_2_3"/><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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Z</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:57_1_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="card_1.html#T21" target="_self">Th21</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:57_1_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Z</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E7:57_1_2"/>
<font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Z</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">m</font> <a href="wellord2.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T8" target="_self">Th8</a>, <a class="ref" href="card_1.html#T22" target="_self">Th22</a>, <a class="ref" href="card_1.html#T23" target="_self">Th23</a>, <a class="txt" href="card_1.html#E3:57_1"><i><font color="Green">E49</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:57_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">Z</font></span><a href="tarski.html#K1">}</a></span>]
 <b>by </b><i><a class="ref" href="card_1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:57"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">X</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(, <a class="txt" href="card_1.html#E3:57"><i><font color="Green">E42</font></i></a>, );</i><br/>


</div>
