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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="zf_refle.html#NM1">Subclass</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="zf_fund1.html#V8" target="_self">closed_wrt_A1-A7</a>
 ;<br/>

<b>assume </b><a NAME="E2:27"/>
not  <a href="ordinal1.html#K5">omega</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then 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="E4:27"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; not <font color="Maroon">c<sub>3</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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="zf_fund1.html#E4:27"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; not <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> );<br/>
<a NAME="E6:27"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &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="zf_fund1.html#E4:27"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E7:27"/><b>then </b><i><font color="Green">E34</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:27"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>(<a class="txt" href="zf_fund1.html#E7:27"><i><font color="Green">E34</font></i></a>);<br/></i>
<a NAME="E10:27"/>
<font color="Maroon">c<sub>5</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="zf_fund1.html#E1:27"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green">E35</font></i></a>, <a class="ref" href="zf_fund1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E11:27"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<a NAME="E12:27"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E13:27"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green">E35</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E39</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:27_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:27_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E13:27"><i><font color="Green">E38</font></i></a>, <a class="txt" href="zf_fund1.html#E1:27_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:27"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E11:27"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E17:27"/><i><font color="Green">E39</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>
<a NAME="E18:27"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E17:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2"/>
not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:27_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zf_fund1.html#E12:27"><i><font color="Green">E37</font></i></a>, <a class="txt" href="zf_fund1.html#E18:27"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E3:27_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E18:27"><i><font color="Green">E40</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><a NAME="E4:27_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:27_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E5:27_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E17:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T14">ORDINAL1:14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E20:27"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:27"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E21:27"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:27"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zf_fund1.html#E19:27"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zf_fund1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E22:27"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E9:27"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zf_fund1.html#E17:27"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#D1">ORDINAL1:def 1</a></i>;<br/>


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