Theorem en2eqpr 8830

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

Step	Hyp	Ref	Expression
1		2onn 7720	. . . . . 6 |- 2o e. om
2		nnfi 8153	. . . . . 6 |- ( 2o e. om -> 2o e. Fin )
3	1, 2	ax-mp 5	. . . . 5 |- 2o e. Fin
4		simpl1 1064	. . . . 5 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> C ~~ 2o )
5		enfii 8177	. . . . 5 |- ( ( 2o e. Fin /\ C ~~ 2o ) -> C e. Fin )
6	3, 4, 5	sylancr 695	. . . 4 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> C e. Fin )
7		simpl2 1065	. . . . 5 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> A e. C )
8		simpl3 1066	. . . . 5 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> B e. C )
9		prssi 4353	. . . . 5 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
10	7, 8, 9	syl2anc 693	. . . 4 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> { A , B } C_ C )
11		pr2nelem 8827	. . . . . . 7 |- ( ( A e. C /\ B e. C /\ A =/= B ) -> { A , B } ~~ 2o )
12	11	3expa 1265	. . . . . 6 |- ( ( ( A e. C /\ B e. C ) /\ A =/= B ) -> { A , B } ~~ 2o )
13	12	3adantl1 1217	. . . . 5 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> { A , B } ~~ 2o )
14	4	ensymd 8007	. . . . 5 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> 2o ~~ C )
15		entr 8008	. . . . 5 |- ( ( { A , B } ~~ 2o /\ 2o ~~ C ) -> { A , B } ~~ C )
16	13, 14, 15	syl2anc 693	. . . 4 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> { A , B } ~~ C )
17		fisseneq 8171	. . . 4 |- ( ( C e. Fin /\ { A , B } C_ C /\ { A , B } ~~ C ) -> { A , B } = C )
18	6, 10, 16, 17	syl3anc 1326	. . 3 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> { A , B } = C )
19	18	eqcomd 2628	. 2 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> C = { A , B } )
20	19	ex 450	1 |- ( ( C ~~ 2o /\ A e. C /\ B e. C ) -> ( A =/= B -> C = { A , B } ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   {cpr 4179   class class class wbr 4653   omcom 7065   2oc2o 7554   ~~ cen 7952   Fincfn 7955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  isprm2lem  15394  en2top  20789