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Theorem en2eqpr 7637
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 6638 . . . . . 6  |-  2o  e.  om
2 nnfi 7053 . . . . . 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 7080 . . . . 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 3771 . . . . 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 7634 . . . . . . 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 6910 . . . . . 6  |-  ( C 
~~  2o  ->  2o  ~~  C )
154, 14syl 15 . . . . 5  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  2o  ~~  C
)
16 entr 6913 . . . . 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 7074 . . . 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 2288 . 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 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   {cpr 3641   class class class wbr 4023   omcom 4656   2oc2o 6473    ~~ cen 6860   Fincfn 6863
This theorem is referenced by:  en2top  16723
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867
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