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<b>let </b><font color="Maroon">c<sub>6</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23"/>
( <font color="Maroon">c<sub>6</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K7">the_axiom_of_pairs</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K8">the_axiom_of_unions</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K9">the_axiom_of_infinity</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K10">the_axiom_of_power_sets</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a> st <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="zf_model.html#R2">|=</a>  <a href="zf_model.html#K11">the_axiom_of_substitution_for</a> <font color="Olive">b<sub>1</sub></font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D12">ZF_MODEL:def 12</a><br/>

<b>hence </b><a NAME="E2:23"/>
( <font color="Maroon">c<sub>6</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> holds <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> holds  <a href="tarski.html#K3">union</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> holds <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>6</sub></font> st <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</a>  <a href="zf_model.html#K2">Free</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="zf_model.html#R1">|=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="zf_lang.html#K12">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span> <a href="zf_lang.html#K4">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>6</sub></font> holds <span class="p1">(<span class="default"><a href="zfmodel1.html#K1" target="_self">def_func'</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) )
 <b>by </b><i><a class="txt" href="zfmodel1.html#E27"><i><font color="Green">Lemma14</font></i></a>, <a class="ref" href="zfmodel1.html#T2" target="_self">Th2</a>, <a class="ref" href="zfmodel1.html#T4" target="_self">Th4</a>, <a class="ref" href="zfmodel1.html#T6" target="_self">Th6</a>, <a class="ref" href="zfmodel1.html#T8" target="_self">Th8</a>, <a class="ref" href="zfmodel1.html#T19" target="_self">Th19</a></i>;<br/>


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