Theorem en2eqpr 6794

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 6634	. . . . . 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 1002	. . . 4 |- ( ( C ~~ 2o /\ A e. C /\ B e. C ) -> E. f f : C -1-1-onto-> 2o )
4	3	adantr 274	. . 3 |- ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) -> E. f f : C -1-1-onto-> 2o )
5		simplr 519	. . . . . . . . . . 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 2173	. . . . . . . . . 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 5359	. . . . . . . . . . . . . 14 |- ( f : C -1-1-onto-> 2o -> f : C -1-1-> 2o )
9	8	adantl 275	. . . . . . . . . . . . 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 274	. . . . . . . . . . . 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 1022	. . . . . . . . . . . . 13 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> B e. C )
13	12	adantr 274	. . . . . . . . . . . 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 5666	. . . . . . . . . . . 12 |- ( ( f : C -1-1-> 2o /\ ( x e. C /\ B e. C ) ) -> ( ( f ` x ) = ( f ` B ) <-> x = B ) )
15	10, 11, 13, 14	syl12anc 1214	. . . . . . . . . . 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 479	. . . . . . . . . 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 3623	. . . . . . . . . . 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 479	. . . . . . . . 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 2214	. . . . . . . 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 523	. . . . . . . . . . . 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 2173	. . . . . . . . . . 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 1021	. . . . . . . . . . . . . 14 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> A e. C )
26	25	adantr 274	. . . . . . . . . . . . 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 5666	. . . . . . . . . . . . 13 |- ( ( f : C -1-1-> 2o /\ ( x e. C /\ A e. C ) ) -> ( ( f ` x ) = ( f ` A ) <-> x = A ) )
28	10, 11, 26, 27	syl12anc 1214	. . . . . . . . . . . 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 483	. . . . . . . . . . 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 3622	. . . . . . . . . . . 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 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 ) = (/) ) -> A e. { A , B } )
34	30, 33	eqeltrd 2214	. . . . . . . . 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 519	. . . . . . . . . . 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 2173	. . . . . . . . . 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 519	. . . . . . . . . . . . 13 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> A =/= B )
39	38	neneqd 2327	. . . . . . . . . . . 12 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> -. A = B )
40		f1fveq 5666	. . . . . . . . . . . . 13 |- ( ( f : C -1-1-> 2o /\ ( A e. C /\ B e. C ) ) -> ( ( f ` A ) = ( f ` B ) <-> A = B ) )
41	9, 25, 12, 40	syl12anc 1214	. . . . . . . . . . . 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 662	. . . . . . . . . . 11 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> -. ( f ` A ) = ( f ` B ) )
43	42	ad4antr 485	. . . . . . . . . 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 609	. . . . . . . . 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 5360	. . . . . . . . . . . . 13 |- ( f : C -1-1-onto-> 2o -> f : C --> 2o )
46	45	adantl 275	. . . . . . . . . . . 12 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> f : C --> 2o )
47	46, 25	ffvelrnd 5549	. . . . . . . . . . 11 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( f ` A ) e. 2o )
48		elpri 3545	. . . . . . . . . . . 12 |- ( ( f ` A ) e. { (/) , 1o } -> ( ( f ` A ) = (/) \/ ( f ` A ) = 1o ) )
49		df2o3 6320	. . . . . . . . . . . 12 |- 2o = { (/) , 1o }
50	48, 49	eleq2s 2232	. . . . . . . . . . 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 483	. . . . . . . . 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 787	. . . . . . . 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 5549	. . . . . . . . . 10 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> ( f ` B ) e. 2o )
55		elpri 3545	. . . . . . . . . . 11 |- ( ( f ` B ) e. { (/) , 1o } -> ( ( f ` B ) = (/) \/ ( f ` B ) = 1o ) )
56	55, 49	eleq2s 2232	. . . . . . . . . 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 479	. . . . . . . 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 787	. . . . . . 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 519	. . . . . . . . . . 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 2173	. . . . . . . . . 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 485	. . . . . . . . . 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 609	. . . . . . . . 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 523	. . . . . . . . . . . 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 2173	. . . . . . . . . . 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 483	. . . . . . . . . . 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 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 ) = 1o ) -> A e. { A , B } )
71	69, 70	eqeltrd 2214	. . . . . . . . 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 483	. . . . . . . . 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 787	. . . . . . . 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 519	. . . . . . . . . . 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 2173	. . . . . . . . . 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 479	. . . . . . . . . 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 479	. . . . . . . . 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 2214	. . . . . . . 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 479	. . . . . . . 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 787	. . . . . . 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 5548	. . . . . . . 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 3545	. . . . . . . . 9 |- ( ( f ` x ) e. { (/) , 1o } -> ( ( f ` x ) = (/) \/ ( f ` x ) = 1o ) )
85	84, 49	eleq2s 2232	. . . . . . . 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 787	. . . . . 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 3098	. . . 4 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> C C_ { A , B } )
90		prssi 3673	. . . . 5 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
91	25, 12, 90	syl2anc 408	. . . 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 3109	. . 3 |- ( ( ( ( C ~~ 2o /\ A e. C /\ B e. C ) /\ A =/= B ) /\ f : C -1-1-onto-> 2o ) -> C = { A , B } )
93	4, 92	exlimddv 1870	. 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 697   /\ w3a 962   = wceq 1331   E.wex 1468   e. wcel 1480   =/= wne 2306   C_ wss 3066   (/)c0 3358   {cpr 3523   class class class wbr 3924   -->wf 5114   -1-1->wf1 5115   -1-1-onto->wf1o 5117   ` cfv 5118   1oc1o 6299   2oc2o 6300   ~~ cen 6625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1o 6306  df-2o 6307  df-en 6628

This theorem is referenced by:  exmidpw  6795  en2eleq  7044  isprm2lem  11786