Theorem en2eqpr 6754

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

Dummy variables f x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6595	. . . . . 6 |- ( C ~~ 2o <-> E. f f : C -1-1-onto-> 2o )
2	1	biimpi 119	. . . . 5 |- ( C ~~ 2o -> E. f f : C -1-1-onto-> 2o )
3	2	3ad2ant1 985	. . . 4 |- ( ( C ~~ 2o /\ A e. C /\ B e. C ) -> E. f f : C -1-1-onto-> 2o )
4	3	adantr 272	. . 3 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> E. f f : C -1-1-onto-> 2o )
5		simplr 502	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = (/) ) -> ( f ` x ) = (/) )
6		simpr 109	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = (/) ) -> ( f ` B ) = (/) )
7	5, 6	eqtr4d 2150	. . . . . . . . . 10 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = (/) ) -> ( f ` x ) = ( f ` B ) )
8		f1of1 5322	. . . . . . . . . . . . . 14 |- ( f : C -1-1-onto-> 2o -> f : C -1-1-> 2o )
9	8	adantl 273	. . . . . . . . . . . . 13 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> f : C -1-1-> 2o )
10	9	adantr 272	. . . . . . . . . . . 12 |- ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) -> f : C -1-1-> 2o )
11		simpr 109	. . . . . . . . . . . 12 |- ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) -> x e. C )
12		simpll3 1005	. . . . . . . . . . . . 13 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> B e. C )
13	12	adantr 272	. . . . . . . . . . . 12 |- ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) -> B e. C )
14		f1fveq 5627	. . . . . . . . . . . 12 |- ( ( f : C -1-1-> 2o /\ ( x e. C /\ B e. C ) ) -> ( ( f ` x ) = ( f ` B ) <-> x = B ) )
15	10, 11, 13, 14	syl12anc 1197	. . . . . . . . . . 11 |- ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) -> ( ( f ` x ) = ( f ` B ) <-> x = B ) )
16	15	ad2antrr 477	. . . . . . . . . 10 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = (/) ) -> ( ( f ` x ) = ( f ` B ) <-> x = B ) )
17	7, 16	mpbid 146	. . . . . . . . 9 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = (/) ) -> x = B )
18		prid2g 3594	. . . . . . . . . . 11 |- ( B e. C -> B e. { A , B } )
19	13, 18	syl 14	. . . . . . . . . 10 |- ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) -> B e. { A , B } )
20	19	ad2antrr 477	. . . . . . . . 9 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = (/) ) -> B e. { A , B } )
21	17, 20	eqeltrd 2191	. . . . . . . 8 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = (/) ) -> x e. { A , B } )
22		simpllr 506	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = (/) ) -> ( f ` x ) = (/) )
23		simpr 109	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = (/) ) -> ( f ` A ) = (/) )
24	22, 23	eqtr4d 2150	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = (/) ) -> ( f ` x ) = ( f ` A ) )
25		simpll2 1004	. . . . . . . . . . . . . 14 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> A e. C )
26	25	adantr 272	. . . . . . . . . . . . 13 |- ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) -> A e. C )
27		f1fveq 5627	. . . . . . . . . . . . 13 |- ( ( f : C -1-1-> 2o /\ ( x e. C /\ A e. C ) ) -> ( ( f ` x ) = ( f ` A ) <-> x = A ) )
28	10, 11, 26, 27	syl12anc 1197	. . . . . . . . . . . 12 |- ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) -> ( ( f ` x ) = ( f ` A ) <-> x = A ) )
29	28	ad3antrrr 481	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = (/) ) -> ( ( f ` x ) = ( f ` A ) <-> x = A ) )
30	24, 29	mpbid 146	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = (/) ) -> x = A )
31		prid1g 3593	. . . . . . . . . . . 12 |- ( A e. C -> A e. { A , B } )
32	26, 31	syl 14	. . . . . . . . . . 11 |- ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) -> A e. { A , B } )
33	32	ad3antrrr 481	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = (/) ) -> A e. { A , B } )
34	30, 33	eqeltrd 2191	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = (/) ) -> x e. { A , B } )
35		simpr 109	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = 1o ) -> ( f ` A ) = 1o )
36		simplr 502	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = 1o ) -> ( f ` B ) = 1o )
37	35, 36	eqtr4d 2150	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = 1o ) -> ( f ` A ) = ( f ` B ) )
38		simplr 502	. . . . . . . . . . . . 13 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> A =/= B )
39	38	neneqd 2303	. . . . . . . . . . . 12 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> -. A = B )
40		f1fveq 5627	. . . . . . . . . . . . 13 |- ( ( f : C -1-1-> 2o /\ ( A e. C /\ B e. C ) ) -> ( ( f ` A ) = ( f ` B ) <-> A = B ) )
41	9, 25, 12, 40	syl12anc 1197	. . . . . . . . . . . 12 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( ( f ` A ) = ( f ` B ) <-> A = B ) )
42	39, 41	mtbird 645	. . . . . . . . . . 11 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> -. ( f ` A ) = ( f ` B ) )
43	42	ad4antr 483	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = 1o ) -> -. ( f ` A ) = ( f ` B ) )
44	37, 43	pm2.21dd 592	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) /\ ( f ` A ) = 1o ) -> x e. { A , B } )
45		f1of 5323	. . . . . . . . . . . . 13 |- ( f : C -1-1-onto-> 2o -> f : C --> 2o )
46	45	adantl 273	. . . . . . . . . . . 12 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> f : C --> 2o )
47	46, 25	ffvelrnd 5510	. . . . . . . . . . 11 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( f ` A ) e. 2o )
48		elpri 3516	. . . . . . . . . . . 12 |- ( ( f ` A ) e. { (/) , 1o } -> ( ( f ` A ) = (/) \/ ( f ` A ) = 1o ) )
49		df2o3 6281	. . . . . . . . . . . 12 |- 2o = { (/) , 1o }
50	48, 49	eleq2s 2209	. . . . . . . . . . 11 |- ( ( f ` A ) e. 2o -> ( ( f ` A ) = (/) \/ ( f ` A ) = 1o ) )
51	47, 50	syl 14	. . . . . . . . . 10 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( ( f ` A ) = (/) \/ ( f ` A ) = 1o ) )
52	51	ad3antrrr 481	. . . . . . . . 9 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) -> ( ( f ` A ) = (/) \/ ( f ` A ) = 1o ) )
53	34, 44, 52	mpjaodan 770	. . . . . . . 8 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) /\ ( f ` B ) = 1o ) -> x e. { A , B } )
54	46, 12	ffvelrnd 5510	. . . . . . . . . 10 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( f ` B ) e. 2o )
55		elpri 3516	. . . . . . . . . . 11 |- ( ( f ` B ) e. { (/) , 1o } -> ( ( f ` B ) = (/) \/ ( f ` B ) = 1o ) )
56	55, 49	eleq2s 2209	. . . . . . . . . 10 |- ( ( f ` B ) e. 2o -> ( ( f ` B ) = (/) \/ ( f ` B ) = 1o ) )
57	54, 56	syl 14	. . . . . . . . 9 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( ( f ` B ) = (/) \/ ( f ` B ) = 1o ) )
58	57	ad2antrr 477	. . . . . . . 8 |- ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) -> ( ( f ` B ) = (/) \/ ( f ` B ) = 1o ) )
59	21, 53, 58	mpjaodan 770	. . . . . . 7 |- ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = (/) ) -> x e. { A , B } )
60		simpr 109	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = (/) ) -> ( f ` A ) = (/) )
61		simplr 502	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = (/) ) -> ( f ` B ) = (/) )
62	60, 61	eqtr4d 2150	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = (/) ) -> ( f ` A ) = ( f ` B ) )
63	42	ad4antr 483	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = (/) ) -> -. ( f ` A ) = ( f ` B ) )
64	62, 63	pm2.21dd 592	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = (/) ) -> x e. { A , B } )
65		simpllr 506	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = 1o ) -> ( f ` x ) = 1o )
66		simpr 109	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = 1o ) -> ( f ` A ) = 1o )
67	65, 66	eqtr4d 2150	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = 1o ) -> ( f ` x ) = ( f ` A ) )
68	28	ad3antrrr 481	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = 1o ) -> ( ( f ` x ) = ( f ` A ) <-> x = A ) )
69	67, 68	mpbid 146	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = 1o ) -> x = A )
70	32	ad3antrrr 481	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = 1o ) -> A e. { A , B } )
71	69, 70	eqeltrd 2191	. . . . . . . . 9 |- ( ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) /\ ( f ` A ) = 1o ) -> x e. { A , B } )
72	51	ad3antrrr 481	. . . . . . . . 9 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) -> ( ( f ` A ) = (/) \/ ( f ` A ) = 1o ) )
73	64, 71, 72	mpjaodan 770	. . . . . . . 8 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = (/) ) -> x e. { A , B } )
74		simplr 502	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = 1o ) -> ( f ` x ) = 1o )
75		simpr 109	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = 1o ) -> ( f ` B ) = 1o )
76	74, 75	eqtr4d 2150	. . . . . . . . . 10 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = 1o ) -> ( f ` x ) = ( f ` B ) )
77	15	ad2antrr 477	. . . . . . . . . 10 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = 1o ) -> ( ( f ` x ) = ( f ` B ) <-> x = B ) )
78	76, 77	mpbid 146	. . . . . . . . 9 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = 1o ) -> x = B )
79	19	ad2antrr 477	. . . . . . . . 9 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = 1o ) -> B e. { A , B } )
80	78, 79	eqeltrd 2191	. . . . . . . 8 |- ( ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) /\ ( f ` B ) = 1o ) -> x e. { A , B } )
81	57	ad2antrr 477	. . . . . . . 8 |- ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) -> ( ( f ` B ) = (/) \/ ( f ` B ) = 1o ) )
82	73, 80, 81	mpjaodan 770	. . . . . . 7 |- ( ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) /\ ( f ` x ) = 1o ) -> x e. { A , B } )
83	46	ffvelrnda 5509	. . . . . . . 8 |- ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) -> ( f ` x ) e. 2o )
84		elpri 3516	. . . . . . . . 9 |- ( ( f ` x ) e. { (/) , 1o } -> ( ( f ` x ) = (/) \/ ( f ` x ) = 1o ) )
85	84, 49	eleq2s 2209	. . . . . . . 8 |- ( ( f ` x ) e. 2o -> ( ( f ` x ) = (/) \/ ( f ` x ) = 1o ) )
86	83, 85	syl 14	. . . . . . 7 |- ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) -> ( ( f ` x ) = (/) \/ ( f ` x ) = 1o ) )
87	59, 82, 86	mpjaodan 770	. . . . . 6 |- ( ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) /\ x e. C ) -> x e. { A , B } )
88	87	ex 114	. . . . 5 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( x e. C -> x e. { A , B } ) )
89	88	ssrdv 3069	. . . 4 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> C C_ { A , B } )
90		prssi 3644	. . . . 5 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
91	25, 12, 90	syl2anc 406	. . . 4 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> { A , B } C_ C )
92	89, 91	eqssd 3080	. . 3 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> C = { A , B } )
93	4, 92	exlimddv 1852	. 2 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> C = { A , B } )
94	93	ex 114	1 |- ( ( C ~~ 2o /\ A e. C /\ B e. C ) -> ( A =/= B -> C = { A , B } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 680   /\ w3a 945   = wceq 1314   E.wex 1451   e. wcel 1463   =/= wne 2282   C_ wss 3037   (/)c0 3329   {cpr 3494   class class class wbr 3895   -->wf 5077   -1-1->wf1 5078   -1-1-onto->wf1o 5080   ` cfv 5081   1oc1o 6260   2oc2o 6261   ~~ cen 6586

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-id 4175  df-suc 4253  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-1o 6267  df-2o 6268  df-en 6589

This theorem is referenced by:  exmidpw  6755  en2eleq  6999  isprm2lem  11643