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Theorem en2eqpr 7653
Description: Building a set with two elements. (Contributed by FL, 11-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
en2eqpr  |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  ->  ( A  =/=  B  ->  C  =  { A ,  B } ) )

Proof of Theorem en2eqpr
StepHypRef Expression
1 2onn 6654 . . . . . 6  |-  2o  e.  om
2 nnfi 7069 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
31, 2ax-mp 8 . . . . 5  |-  2o  e.  Fin
4 simpl1 958 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  ~~  2o )
5 enfii 7096 . . . . 5  |-  ( ( 2o  e.  Fin  /\  C  ~~  2o )  ->  C  e.  Fin )
63, 4, 5sylancr 644 . . . 4  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  e.  Fin )
7 simpl2 959 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  A  e.  C )
8 simpl3 960 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  B  e.  C )
9 prssi 3787 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
107, 8, 9syl2anc 642 . . . 4  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  C_  C
)
11 pr2nelem 7650 . . . . . . 7  |-  ( ( A  e.  C  /\  B  e.  C  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
12113expa 1151 . . . . . 6  |-  ( ( ( A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
13123adantl1 1111 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  ~~  2o )
14 ensym 6926 . . . . . 6  |-  ( C 
~~  2o  ->  2o  ~~  C )
154, 14syl 15 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  2o  ~~  C
)
16 entr 6929 . . . . 5  |-  ( ( { A ,  B }  ~~  2o  /\  2o  ~~  C )  ->  { A ,  B }  ~~  C
)
1713, 15, 16syl2anc 642 . . . 4  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  ~~  C
)
18 fisseneq 7090 . . . 4  |-  ( ( C  e.  Fin  /\  { A ,  B }  C_  C  /\  { A ,  B }  ~~  C
)  ->  { A ,  B }  =  C )
196, 10, 17, 18syl3anc 1182 . . 3  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  =  C )
2019eqcomd 2301 . 2  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  =  { A ,  B }
)
2120ex 423 1  |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  ->  ( A  =/=  B  ->  C  =  { A ,  B } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   {cpr 3654   class class class wbr 4039   omcom 4672   2oc2o 6489    ~~ cen 6876   Fincfn 6879
This theorem is referenced by:  en2top  16739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-2o 6496  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883
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