Theorem en2eqpr 7098

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 6916	. . . . . 6 |- ( C ~~ 2o <-> E. f f : C -1-1-onto-> 2o )
2	1	biimpi 120	. . . . 5 |- ( C ~~ 2o -> E. f f : C -1-1-onto-> 2o )
3	2	3ad2ant1 1044	. . . 4 |- ( ( C ~~ 2o /\ A e. C /\ B e. C ) -> E. f f : C -1-1-onto-> 2o )
4	3	adantr 276	. . 3 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> E. f f : C -1-1-onto-> 2o )
5		simplr 529	. . . . . . . . . . 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 110	. . . . . . . . . . 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 2267	. . . . . . . . . 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 5582	. . . . . . . . . . . . . 14 |- ( f : C -1-1-onto-> 2o -> f : C -1-1-> 2o )
9	8	adantl 277	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 110	. . . . . . . . . . . 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 1064	. . . . . . . . . . . . 13 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> B e. C )
13	12	adantr 276	. . . . . . . . . . . 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 5912	. . . . . . . . . . . 12 |- ( ( f : C -1-1-> 2o /\ ( x e. C /\ B e. C ) ) -> ( ( f ` x ) = ( f ` B ) <-> x = B ) )
15	10, 11, 13, 14	syl12anc 1271	. . . . . . . . . . 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 488	. . . . . . . . . 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 147	. . . . . . . . 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 3776	. . . . . . . . . . 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 488	. . . . . . . . 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 2308	. . . . . . . 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 536	. . . . . . . . . . . 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 110	. . . . . . . . . . . 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 2267	. . . . . . . . . . 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 1063	. . . . . . . . . . . . . 14 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> A e. C )
26	25	adantr 276	. . . . . . . . . . . . 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 5912	. . . . . . . . . . . . 13 |- ( ( f : C -1-1-> 2o /\ ( x e. C /\ A e. C ) ) -> ( ( f ` x ) = ( f ` A ) <-> x = A ) )
28	10, 11, 26, 27	syl12anc 1271	. . . . . . . . . . . 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 492	. . . . . . . . . . 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 147	. . . . . . . . . 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 3775	. . . . . . . . . . . 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 492	. . . . . . . . . 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 2308	. . . . . . . . 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 110	. . . . . . . . . . 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 529	. . . . . . . . . . 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 2267	. . . . . . . . . 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 529	. . . . . . . . . . . . 13 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> A =/= B )
39	38	neneqd 2423	. . . . . . . . . . . 12 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> -. A = B )
40		f1fveq 5912	. . . . . . . . . . . . 13 |- ( ( f : C -1-1-> 2o /\ ( A e. C /\ B e. C ) ) -> ( ( f ` A ) = ( f ` B ) <-> A = B ) )
41	9, 25, 12, 40	syl12anc 1271	. . . . . . . . . . . 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 679	. . . . . . . . . . 11 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> -. ( f ` A ) = ( f ` B ) )
43	42	ad4antr 494	. . . . . . . . . 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 625	. . . . . . . . 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 5583	. . . . . . . . . . . . 13 |- ( f : C -1-1-onto-> 2o -> f : C --> 2o )
46	45	adantl 277	. . . . . . . . . . . 12 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> f : C --> 2o )
47	46, 25	ffvelcdmd 5783	. . . . . . . . . . 11 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( f ` A ) e. 2o )
48		elpri 3692	. . . . . . . . . . . 12 |- ( ( f ` A ) e. { (/) , 1o } -> ( ( f ` A ) = (/) \/ ( f ` A ) = 1o ) )
49		df2o3 6596	. . . . . . . . . . . 12 |- 2o = { (/) , 1o }
50	48, 49	eleq2s 2326	. . . . . . . . . . 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 492	. . . . . . . . 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 805	. . . . . . . 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	ffvelcdmd 5783	. . . . . . . . . 10 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( f ` B ) e. 2o )
55		elpri 3692	. . . . . . . . . . 11 |- ( ( f ` B ) e. { (/) , 1o } -> ( ( f ` B ) = (/) \/ ( f ` B ) = 1o ) )
56	55, 49	eleq2s 2326	. . . . . . . . . 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 488	. . . . . . . 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 805	. . . . . . 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 110	. . . . . . . . . . 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 529	. . . . . . . . . . 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 2267	. . . . . . . . . 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 494	. . . . . . . . . 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 625	. . . . . . . . 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 536	. . . . . . . . . . . 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 110	. . . . . . . . . . . 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 2267	. . . . . . . . . . 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 492	. . . . . . . . . . 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 147	. . . . . . . . . 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 492	. . . . . . . . . 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 2308	. . . . . . . . 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 492	. . . . . . . . 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 805	. . . . . . . 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 529	. . . . . . . . . . 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 110	. . . . . . . . . . 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 2267	. . . . . . . . . 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 488	. . . . . . . . . 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 147	. . . . . . . . 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 488	. . . . . . . . 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 2308	. . . . . . . 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 488	. . . . . . . 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 805	. . . . . . 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	ffvelcdmda 5782	. . . . . . . 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 3692	. . . . . . . . 9 |- ( ( f ` x ) e. { (/) , 1o } -> ( ( f ` x ) = (/) \/ ( f ` x ) = 1o ) )
85	84, 49	eleq2s 2326	. . . . . . . 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 805	. . . . . 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 115	. . . . 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 3233	. . . 4 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> C C_ { A , B } )
90		prssi 3831	. . . . 5 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
91	25, 12, 90	syl2anc 411	. . . 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 3244	. . 3 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> C = { A , B } )
93	4, 92	exlimddv 1947	. 2 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> C = { A , B } )
94	93	ex 115	1 |- ( ( C ~~ 2o /\ A e. C /\ B e. C ) -> ( A =/= B -> C = { A , B } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   =/= wne 2402   C_ wss 3200   (/)c0 3494   {cpr 3670   class class class wbr 4088   -->wf 5322   -1-1->wf1 5323   -1-1-onto->wf1o 5325   ` cfv 5326   1oc1o 6574   2oc2o 6575   ~~ cen 6906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-2o 6582  df-en 6909

This theorem is referenced by:  exmidpw  7099  en2eleq  7405  isprm2lem  12687