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Theorem en2eqpr 8381
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 7286 . . . . . 6  |-  2o  e.  om
2 nnfi 7707 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
31, 2ax-mp 5 . . . . 5  |-  2o  e.  Fin
4 simpl1 999 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  ~~  2o )
5 enfii 7734 . . . . 5  |-  ( ( 2o  e.  Fin  /\  C  ~~  2o )  ->  C  e.  Fin )
63, 4, 5sylancr 663 . . . 4  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  e.  Fin )
7 simpl2 1000 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  A  e.  C )
8 simpl3 1001 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  B  e.  C )
9 prssi 4183 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
107, 8, 9syl2anc 661 . . . 4  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  C_  C
)
11 pr2nelem 8378 . . . . . . 7  |-  ( ( A  e.  C  /\  B  e.  C  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
12113expa 1196 . . . . . 6  |-  ( ( ( A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
13123adantl1 1152 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  ~~  2o )
144ensymd 7563 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  2o  ~~  C
)
15 entr 7564 . . . . 5  |-  ( ( { A ,  B }  ~~  2o  /\  2o  ~~  C )  ->  { A ,  B }  ~~  C
)
1613, 14, 15syl2anc 661 . . . 4  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  ~~  C
)
17 fisseneq 7728 . . . 4  |-  ( ( C  e.  Fin  /\  { A ,  B }  C_  C  /\  { A ,  B }  ~~  C
)  ->  { A ,  B }  =  C )
186, 10, 16, 17syl3anc 1228 . . 3  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  =  C )
1918eqcomd 2475 . 2  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  =  { A ,  B }
)
2019ex 434 1  |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  ->  ( A  =/=  B  ->  C  =  { A ,  B } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3476   {cpr 4029   class class class wbr 4447   omcom 6678   2oc2o 7121    ~~ cen 7510   Fincfn 7513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-1o 7127  df-2o 7128  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517
This theorem is referenced by:  en2top  19253
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