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Theorem en2eqpr 8280
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 7184 . . . . . 6  |-  2o  e.  om
2 nnfi 7609 . . . . . 6  |-  ( 2o  e.  om  ->  2o  e.  Fin )
31, 2ax-mp 5 . . . . 5  |-  2o  e.  Fin
4 simpl1 991 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  ~~  2o )
5 enfii 7636 . . . . 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 992 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  A  e.  C )
8 simpl3 993 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  B  e.  C )
9 prssi 4132 . . . . 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 8277 . . . . . . 7  |-  ( ( A  e.  C  /\  B  e.  C  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
12113expa 1188 . . . . . 6  |-  ( ( ( A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  ->  { A ,  B }  ~~  2o )
13123adantl1 1144 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  ~~  2o )
144ensymd 7465 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  2o  ~~  C
)
15 entr 7466 . . . . 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 7630 . . . 4  |-  ( ( C  e.  Fin  /\  { A ,  B }  C_  C  /\  { A ,  B }  ~~  C
)  ->  { A ,  B }  =  C )
186, 10, 16, 17syl3anc 1219 . . 3  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  { A ,  B }  =  C )
1918eqcomd 2460 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2645    C_ wss 3431   {cpr 3982   class class class wbr 4395   omcom 6581   2oc2o 7019    ~~ cen 7412   Fincfn 7415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-om 6582  df-1o 7025  df-2o 7026  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419
This theorem is referenced by:  en2top  18717
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