Theorem en2eqpr 7204

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 7020	. . . . . 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 1049	. . . 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 533	. . . . . . . . . . 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 2274	. . . . . . . . . 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 5633	. . . . . . . . . . . . . 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 1069	. . . . . . . . . . . . 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 5968	. . . . . . . . . . . 12 |- ( ( f : C -1-1-> 2o /\ ( x e. C /\ B e. C ) ) -> ( ( f ` x ) = ( f ` B ) <-> x = B ) )
15	10, 11, 13, 14	syl12anc 1276	. . . . . . . . . . 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 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 ) = (/) ) -> ( ( 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 3812	. . . . . . . . . . 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 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 ) = (/) ) -> B e. { A , B } )
21	17, 20	eqeltrd 2315	. . . . . . . 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 540	. . . . . . . . . . . 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 2274	. . . . . . . . . . 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 1068	. . . . . . . . . . . . . 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 5968	. . . . . . . . . . . . 13 |- ( ( f : C -1-1-> 2o /\ ( x e. C /\ A e. C ) ) -> ( ( f ` x ) = ( f ` A ) <-> x = A ) )
28	10, 11, 26, 27	syl12anc 1276	. . . . . . . . . . . 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 496	. . . . . . . . . . 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 3811	. . . . . . . . . . . 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 496	. . . . . . . . . 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 2315	. . . . . . . . 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 533	. . . . . . . . . . 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 2274	. . . . . . . . . 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 533	. . . . . . . . . . . . 13 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> A =/= B )
39	38	neneqd 2441	. . . . . . . . . . . 12 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> -. A = B )
40		f1fveq 5968	. . . . . . . . . . . . 13 |- ( ( f : C -1-1-> 2o /\ ( A e. C /\ B e. C ) ) -> ( ( f ` A ) = ( f ` B ) <-> A = B ) )
41	9, 25, 12, 40	syl12anc 1276	. . . . . . . . . . . 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 684	. . . . . . . . . . 11 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> -. ( f ` A ) = ( f ` B ) )
43	42	ad4antr 498	. . . . . . . . . 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 629	. . . . . . . . 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 5634	. . . . . . . . . . . . 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 5835	. . . . . . . . . . 11 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( f ` A ) e. 2o )
48		elpri 3728	. . . . . . . . . . . 12 |- ( ( f ` A ) e. { (/) , 1o } -> ( ( f ` A ) = (/) \/ ( f ` A ) = 1o ) )
49		df2o3 6692	. . . . . . . . . . . 12 |- 2o = { (/) , 1o }
50	48, 49	eleq2s 2333	. . . . . . . . . . 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 496	. . . . . . . . 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 810	. . . . . . . 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 5835	. . . . . . . . . 10 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( f ` B ) e. 2o )
55		elpri 3728	. . . . . . . . . . 11 |- ( ( f ` B ) e. { (/) , 1o } -> ( ( f ` B ) = (/) \/ ( f ` B ) = 1o ) )
56	55, 49	eleq2s 2333	. . . . . . . . . 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 492	. . . . . . . 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 810	. . . . . . 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 533	. . . . . . . . . . 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 2274	. . . . . . . . . 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 498	. . . . . . . . . 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 629	. . . . . . . . 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 540	. . . . . . . . . . . 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 2274	. . . . . . . . . . 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 496	. . . . . . . . . . 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 496	. . . . . . . . . 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 2315	. . . . . . . . 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 496	. . . . . . . . 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 810	. . . . . . . 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 533	. . . . . . . . . . 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 2274	. . . . . . . . . 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 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 ) = 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 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 ) = 1o ) -> B e. { A , B } )
80	78, 79	eqeltrd 2315	. . . . . . . 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 492	. . . . . . . 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 810	. . . . . . 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 5834	. . . . . . . 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 3728	. . . . . . . . 9 |- ( ( f ` x ) e. { (/) , 1o } -> ( ( f ` x ) = (/) \/ ( f ` x ) = 1o ) )
85	84, 49	eleq2s 2333	. . . . . . . 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 810	. . . . . 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 3254	. . . 4 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> C C_ { A , B } )
90		prssi 3868	. . . . 5 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
91	25, 12, 90	syl2anc 415	. . . 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 3265	. . 3 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> C = { A , B } )
93	4, 92	exlimddv 1954	. 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 720   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   =/= wne 2420   C_ wss 3220   (/)c0 3520   {cpr 3706   class class class wbr 4125   -->wf 5368   -1-1->wf1 5369   -1-1-onto->wf1o 5371   ` cfv 5372   1oc1o 6670   2oc2o 6671   ~~ cen 7010

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1o 6677  df-2o 6678  df-en 7013

This theorem is referenced by:  exmidpw  7205  en2eleq  7537  isprm2lem  12872