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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">M</font> <a href="zfrefle1.html#R1" title="ZFREFLE1:pred.1">|=</a>  <a href="zfrefle1.html#K2" title="ZFREFLE1:func.2">ZF-axioms</a>  &amp; <font color="Maroon" title="c1">M</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  implies <font color="Maroon" title="c1">M</font> is  <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a>  )</span><br/>

<a NAME="E1:9"/>
 <a href="zf_model.html#K10" title="ZF_MODEL:func.10">the_axiom_of_power_sets</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</a></span>;<br/>
<a NAME="E2:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A1</font></span>: 
 <a href="zf_model.html#K10" title="ZF_MODEL:func.10">the_axiom_of_power_sets</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfrefle1.html#K2" title="ZFREFLE1:func.2">ZF-axioms</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfrefle1.html#D4" target="_self" title="ZFREFLE1:def.4">Def4</a></span>;<br/>
<a NAME="E3:9"/>
 <a href="zf_model.html#K9" title="ZF_MODEL:func.9">the_axiom_of_infinity</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</a></span>;<br/>
<a NAME="E4:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A2</font></span>: 
 <a href="zf_model.html#K9" title="ZF_MODEL:func.9">the_axiom_of_infinity</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfrefle1.html#K2" title="ZFREFLE1:func.2">ZF-axioms</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfrefle1.html#D4" target="_self" title="ZFREFLE1:def.4">Def4</a></span>;<br/>
<a NAME="E5:9"/>
 <a href="zf_model.html#K8" title="ZF_MODEL:func.8">the_axiom_of_unions</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</a></span>;<br/>
<a NAME="E6:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 <a href="zf_model.html#K8" title="ZF_MODEL:func.8">the_axiom_of_unions</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfrefle1.html#K2" title="ZFREFLE1:func.2">ZF-axioms</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfrefle1.html#D4" target="_self" title="ZFREFLE1:def.4">Def4</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:9"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfrefle1.html#K2" title="ZFREFLE1:func.2">ZF-axioms</a>  holds <br/><font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a> <font color="Olive" title="b1">H</font>
 <span class="kw">and </span><br/><a NAME="E8:9"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
<font color="Maroon" title="c1">M</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="zfrefle1.html#D1" target="_self" title="ZFREFLE1:def.1">ZFREFLE1:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font> is  <a href="zf_model.html#V1" title="ZF_MODEL:attr.1">being_a_model_of_ZF</a> </span><br/>

<a NAME="E9:9"/>
 <a href="zf_model.html#K7" title="ZF_MODEL:func.7">the_axiom_of_pairs</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</a></span>;<br/>
<a NAME="E10:9"/><span class="kw">then </span>
 <a href="zf_model.html#K7" title="ZF_MODEL:func.7">the_axiom_of_pairs</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfrefle1.html#K2" title="ZFREFLE1:func.2">ZF-axioms</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfrefle1.html#D4" target="_self" title="ZFREFLE1:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:9"/>
( <font color="Maroon" title="c1">M</font> is  <a href="ordinal1.html#V1" title="ORDINAL1:attr.1">epsilon-transitive</a>  &amp; <font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K7" title="ZF_MODEL:func.7">the_axiom_of_pairs</a>  &amp; <font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K8" title="ZF_MODEL:func.8">the_axiom_of_unions</a>  &amp; <font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K9" title="ZF_MODEL:func.9">the_axiom_of_infinity</a>  &amp; <font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K10" title="ZF_MODEL:func.10">the_axiom_of_power_sets</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="txt" href="#E8:9"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E7">A1</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="zf_model.html#D12" title="ZF_MODEL:def.12">ZF_MODEL:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">M7</a>( <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> )  holds <br/>(  not <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font> or <font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Maroon" title="c2">H</font> )</span><br/>

<span class="kw">assume </span><a NAME="E12:9"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 2</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Maroon" title="c2">H</font></span><br/>

<a NAME="E13:9"/>
 <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</a></span>;<br/>
<a NAME="E14:9"/><span class="kw">then </span>
 <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfrefle1.html#K2" title="ZFREFLE1:func.2">ZF-axioms</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:9"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="zfrefle1.html#D4" target="_self" title="ZFREFLE1:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:9"/>
<font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_model.html#K11" title="ZF_MODEL:func.11">the_axiom_of_substitution_for</a> <font color="Maroon" title="c2">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E10">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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