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Theorem en2eqpr 6458
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.
StepHypRef Expression
1 bren 6315 . . . . . 6  |-  ( C 
~~  2o  <->  E. f  f : C -1-1-onto-> 2o )
21biimpi 118 . . . . 5  |-  ( C 
~~  2o  ->  E. f 
f : C -1-1-onto-> 2o )
323ad2ant1 960 . . . 4  |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  ->  E. f  f : C -1-1-onto-> 2o )
43adantr 270 . . 3  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  E. f 
f : C -1-1-onto-> 2o )
5 simplr 497 . . . . . . . . . . 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 108 . . . . . . . . . . 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 )  =  (/) )
75, 6eqtr4d 2118 . . . . . . . . . 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 5176 . . . . . . . . . . . . . 14  |-  ( f : C -1-1-onto-> 2o  ->  f : C -1-1-> 2o )
98adantl 271 . . . . . . . . . . . . 13  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  f : C -1-1-> 2o )
109adantr 270 . . . . . . . . . . . 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 108 . . . . . . . . . . . 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 980 . . . . . . . . . . . . 13  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  B  e.  C )
1312adantr 270 . . . . . . . . . . . 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 5463 . . . . . . . . . . . 12  |-  ( ( f : C -1-1-> 2o  /\  ( x  e.  C  /\  B  e.  C
) )  ->  (
( f `  x
)  =  ( f `
 B )  <->  x  =  B ) )
1510, 11, 13, 14syl12anc 1168 . . . . . . . . . . 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 ) )
1615ad2antrr 472 . . . . . . . . . 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 ) )
177, 16mpbid 145 . . . . . . . . 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 3515 . . . . . . . . . . 11  |-  ( B  e.  C  ->  B  e.  { A ,  B } )
1913, 18syl 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 } )
2019ad2antrr 472 . . . . . . . . 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 } )
2117, 20eqeltrd 2159 . . . . . . . 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 501 . . . . . . . . . . . 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 108 . . . . . . . . . . . 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 )  =  (/) )
2422, 23eqtr4d 2118 . . . . . . . . . . 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 979 . . . . . . . . . . . . . 14  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  A  e.  C )
2625adantr 270 . . . . . . . . . . . . 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 5463 . . . . . . . . . . . . 13  |-  ( ( f : C -1-1-> 2o  /\  ( x  e.  C  /\  A  e.  C
) )  ->  (
( f `  x
)  =  ( f `
 A )  <->  x  =  A ) )
2810, 11, 26, 27syl12anc 1168 . . . . . . . . . . . 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 ) )
2928ad3antrrr 476 . . . . . . . . . . 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 ) )
3024, 29mpbid 145 . . . . . . . . . 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 3514 . . . . . . . . . . . 12  |-  ( A  e.  C  ->  A  e.  { A ,  B } )
3226, 31syl 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 } )
3332ad3antrrr 476 . . . . . . . . . 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 } )
3430, 33eqeltrd 2159 . . . . . . . . 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 108 . . . . . . . . . . 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 497 . . . . . . . . . . 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 )
3735, 36eqtr4d 2118 . . . . . . . . . 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 497 . . . . . . . . . . . . 13  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  A  =/=  B )
3938neneqd 2270 . . . . . . . . . . . 12  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  -.  A  =  B )
40 f1fveq 5463 . . . . . . . . . . . . 13  |-  ( ( f : C -1-1-> 2o  /\  ( A  e.  C  /\  B  e.  C
) )  ->  (
( f `  A
)  =  ( f `
 B )  <->  A  =  B ) )
419, 25, 12, 40syl12anc 1168 . . . . . . . . . . . 12  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
( f `  A
)  =  ( f `
 B )  <->  A  =  B ) )
4239, 41mtbird 631 . . . . . . . . . . 11  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  -.  ( f `  A
)  =  ( f `
 B ) )
4342ad4antr 478 . . . . . . . . . 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 ) )
4437, 43pm2.21dd 583 . . . . . . . . 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 5177 . . . . . . . . . . . . 13  |-  ( f : C -1-1-onto-> 2o  ->  f : C
--> 2o )
4645adantl 271 . . . . . . . . . . . 12  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  f : C --> 2o )
4746, 25ffvelrnd 5355 . . . . . . . . . . 11  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
f `  A )  e.  2o )
48 elpri 3439 . . . . . . . . . . . 12  |-  ( ( f `  A )  e.  { (/) ,  1o }  ->  ( ( f `
 A )  =  (/)  \/  ( f `  A )  =  1o ) )
49 df2o3 6098 . . . . . . . . . . . 12  |-  2o  =  { (/) ,  1o }
5048, 49eleq2s 2177 . . . . . . . . . . 11  |-  ( ( f `  A )  e.  2o  ->  (
( f `  A
)  =  (/)  \/  (
f `  A )  =  1o ) )
5147, 50syl 14 . . . . . . . . . 10  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
( f `  A
)  =  (/)  \/  (
f `  A )  =  1o ) )
5251ad3antrrr 476 . . . . . . . . 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 ) )
5334, 44, 52mpjaodan 745 . . . . . . . 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 } )
5446, 12ffvelrnd 5355 . . . . . . . . . 10  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
f `  B )  e.  2o )
55 elpri 3439 . . . . . . . . . . 11  |-  ( ( f `  B )  e.  { (/) ,  1o }  ->  ( ( f `
 B )  =  (/)  \/  ( f `  B )  =  1o ) )
5655, 49eleq2s 2177 . . . . . . . . . 10  |-  ( ( f `  B )  e.  2o  ->  (
( f `  B
)  =  (/)  \/  (
f `  B )  =  1o ) )
5754, 56syl 14 . . . . . . . . 9  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
( f `  B
)  =  (/)  \/  (
f `  B )  =  1o ) )
5857ad2antrr 472 . . . . . . . 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 ) )
5921, 53, 58mpjaodan 745 . . . . . . 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 108 . . . . . . . . . . 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 497 . . . . . . . . . . 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 )  =  (/) )
6260, 61eqtr4d 2118 . . . . . . . . . 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 ) )
6342ad4antr 478 . . . . . . . . . 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 ) )
6462, 63pm2.21dd 583 . . . . . . . . 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 501 . . . . . . . . . . . 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 108 . . . . . . . . . . . 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 )
6765, 66eqtr4d 2118 . . . . . . . . . . 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 ) )
6828ad3antrrr 476 . . . . . . . . . . 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 ) )
6967, 68mpbid 145 . . . . . . . . . 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 )
7032ad3antrrr 476 . . . . . . . . . 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 } )
7169, 70eqeltrd 2159 . . . . . . . . 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 } )
7251ad3antrrr 476 . . . . . . . . 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 ) )
7364, 71, 72mpjaodan 745 . . . . . . . 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 497 . . . . . . . . . . 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 108 . . . . . . . . . . 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 )
7674, 75eqtr4d 2118 . . . . . . . . . 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 ) )
7715ad2antrr 472 . . . . . . . . . 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 ) )
7876, 77mpbid 145 . . . . . . . . 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 )
7919ad2antrr 472 . . . . . . . . 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 } )
8078, 79eqeltrd 2159 . . . . . . . 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 } )
8157ad2antrr 472 . . . . . . . 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 ) )
8273, 80, 81mpjaodan 745 . . . . . . 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 } )
8346ffvelrnda 5354 . . . . . . . 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 3439 . . . . . . . . 9  |-  ( ( f `  x )  e.  { (/) ,  1o }  ->  ( ( f `
 x )  =  (/)  \/  ( f `  x )  =  1o ) )
8584, 49eleq2s 2177 . . . . . . . 8  |-  ( ( f `  x )  e.  2o  ->  (
( f `  x
)  =  (/)  \/  (
f `  x )  =  1o ) )
8683, 85syl 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 ) )
8759, 82, 86mpjaodan 745 . . . . . 6  |-  ( ( ( ( ( C 
~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  /\  x  e.  C )  ->  x  e.  { A ,  B } )
8887ex 113 . . . . 5  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
x  e.  C  ->  x  e.  { A ,  B } ) )
8988ssrdv 3014 . . . 4  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  C  C_ 
{ A ,  B } )
90 prssi 3563 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
9125, 12, 90syl2anc 403 . . . 4  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  { A ,  B }  C_  C
)
9289, 91eqssd 3025 . . 3  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  C  =  { A ,  B } )
934, 92exlimddv 1821 . 2  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  =  { A ,  B }
)
9493ex 113 1  |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  ->  ( A  =/=  B  ->  C  =  { A ,  B } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434    =/= wne 2249    C_ wss 2982   (/)c0 3267   {cpr 3417   class class class wbr 3805   -->wf 4948   -1-1->wf1 4949   -1-1-onto->wf1o 4951   ` cfv 4952   1oc1o 6078   2oc2o 6079    ~~ cen 6306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-1o 6085  df-2o 6086  df-en 6309
This theorem is referenced by:  en2eleq  6573  isprm2lem  10705
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