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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:66"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="zfmisc_1.html#K1">bool</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R2">are_equipotent</a>  or <font color="Olive">b<sub>1</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="zfmisc_1.html#T136">ZFMISC_1:136</a></i>;<br/>
<a NAME="E3:66"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3" target="_self">On</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K3" target="_self">On</a> <font color="Maroon">c<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_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> ;<br/>

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3" target="_self">On</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:66_1"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#D10" target="_self">Def10</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="ordinal1.html#E1:66_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#D10" target="_self">Def10</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</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="E4:66_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E5:66_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E7:66_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <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="ordinal1.html#E1:66_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="ordinal1.html#E4:66_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal1.html#D2" target="_self">Def2</a>, <a class="ref" href="ordinal1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E8:66_1"/><b>then </b>
<font color="Maroon">c<sub>6</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="ordinal1.html#E2:66"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:66_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3" target="_self">On</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D10" target="_self">Def10</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="ordinal1.html#K3" target="_self">On</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T31" target="_self">Th31</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E5:66"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E2:66"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal1.html#D10" target="_self">Def10</a></i>;<br/>

<a NAME="E6:66"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/> <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:66_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:66_2"/><b>then </b>
<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="ref" href="ordinal1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E3:66_2"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="zfmisc_1.html#K1">bool</a> <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="ordinal1.html#E2:66"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E4:66_2"/>
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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="E1:66_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:66_2_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E3:66_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E4:66_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:66_2"/><b>then </b>
 <a href="ordinal1.html#K1" target="_self">succ</a> <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="ordinal1.html#E2:66"><i><font color="Green">E34</font></i></a>, <a class="txt" href="ordinal1.html#E3:66_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:66_2"/>
 <a href="ordinal1.html#K1" target="_self">succ</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D10" target="_self">Def10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:66"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T41" target="_self">Th41</a></i>;<br/>


</div>
