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Theorem en2eqpr 6963
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 6801 . . . . . 6  |-  ( C 
~~  2o  <->  E. f  f : C -1-1-onto-> 2o )
21biimpi 120 . . . . 5  |-  ( C 
~~  2o  ->  E. f 
f : C -1-1-onto-> 2o )
323ad2ant1 1020 . . . 4  |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  ->  E. f  f : C -1-1-onto-> 2o )
43adantr 276 . . 3  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  E. f 
f : C -1-1-onto-> 2o )
5 simplr 528 . . . . . . . . . . 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 )  =  (/) )
75, 6eqtr4d 2229 . . . . . . . . . 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 5499 . . . . . . . . . . . . . 14  |-  ( f : C -1-1-onto-> 2o  ->  f : C -1-1-> 2o )
98adantl 277 . . . . . . . . . . . . 13  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  f : C -1-1-> 2o )
109adantr 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 1040 . . . . . . . . . . . . 13  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  B  e.  C )
1312adantr 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 5815 . . . . . . . . . . . 12  |-  ( ( f : C -1-1-> 2o  /\  ( x  e.  C  /\  B  e.  C
) )  ->  (
( f `  x
)  =  ( f `
 B )  <->  x  =  B ) )
1510, 11, 13, 14syl12anc 1247 . . . . . . . . . . 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 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 ) )
177, 16mpbid 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 3723 . . . . . . . . . . 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 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 } )
2117, 20eqeltrd 2270 . . . . . . . 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 534 . . . . . . . . . . . 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 )  =  (/) )
2422, 23eqtr4d 2229 . . . . . . . . . . 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 1039 . . . . . . . . . . . . . 14  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  A  e.  C )
2625adantr 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 5815 . . . . . . . . . . . . 13  |-  ( ( f : C -1-1-> 2o  /\  ( x  e.  C  /\  A  e.  C
) )  ->  (
( f `  x
)  =  ( f `
 A )  <->  x  =  A ) )
2810, 11, 26, 27syl12anc 1247 . . . . . . . . . . . 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 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 ) )
3024, 29mpbid 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 3722 . . . . . . . . . . . 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 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 } )
3430, 33eqeltrd 2270 . . . . . . . . 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 528 . . . . . . . . . . 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 2229 . . . . . . . . . 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 528 . . . . . . . . . . . . 13  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  A  =/=  B )
3938neneqd 2385 . . . . . . . . . . . 12  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  -.  A  =  B )
40 f1fveq 5815 . . . . . . . . . . . . 13  |-  ( ( f : C -1-1-> 2o  /\  ( A  e.  C  /\  B  e.  C
) )  ->  (
( f `  A
)  =  ( f `
 B )  <->  A  =  B ) )
419, 25, 12, 40syl12anc 1247 . . . . . . . . . . . 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 674 . . . . . . . . . . 11  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  -.  ( f `  A
)  =  ( f `
 B ) )
4342ad4antr 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 ) )
4437, 43pm2.21dd 621 . . . . . . . . 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 5500 . . . . . . . . . . . . 13  |-  ( f : C -1-1-onto-> 2o  ->  f : C
--> 2o )
4645adantl 277 . . . . . . . . . . . 12  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  f : C --> 2o )
4746, 25ffvelcdmd 5694 . . . . . . . . . . 11  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
f `  A )  e.  2o )
48 elpri 3641 . . . . . . . . . . . 12  |-  ( ( f `  A )  e.  { (/) ,  1o }  ->  ( ( f `
 A )  =  (/)  \/  ( f `  A )  =  1o ) )
49 df2o3 6483 . . . . . . . . . . . 12  |-  2o  =  { (/) ,  1o }
5048, 49eleq2s 2288 . . . . . . . . . . 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 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 ) )
5334, 44, 52mpjaodan 799 . . . . . . . 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, 12ffvelcdmd 5694 . . . . . . . . . 10  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
f `  B )  e.  2o )
55 elpri 3641 . . . . . . . . . . 11  |-  ( ( f `  B )  e.  { (/) ,  1o }  ->  ( ( f `
 B )  =  (/)  \/  ( f `  B )  =  1o ) )
5655, 49eleq2s 2288 . . . . . . . . . 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 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 ) )
5921, 53, 58mpjaodan 799 . . . . . . 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 528 . . . . . . . . . . 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 2229 . . . . . . . . . 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 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 ) )
6462, 63pm2.21dd 621 . . . . . . . . 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 534 . . . . . . . . . . . 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 )
6765, 66eqtr4d 2229 . . . . . . . . . . 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 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 ) )
6967, 68mpbid 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 )
7032ad3antrrr 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 } )
7169, 70eqeltrd 2270 . . . . . . . . 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 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 ) )
7364, 71, 72mpjaodan 799 . . . . . . . 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 528 . . . . . . . . . . 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 )
7674, 75eqtr4d 2229 . . . . . . . . . 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 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 ) )
7876, 77mpbid 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 )
7919ad2antrr 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 } )
8078, 79eqeltrd 2270 . . . . . . . 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 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 ) )
8273, 80, 81mpjaodan 799 . . . . . . 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 } )
8346ffvelcdmda 5693 . . . . . . . 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 3641 . . . . . . . . 9  |-  ( ( f `  x )  e.  { (/) ,  1o }  ->  ( ( f `
 x )  =  (/)  \/  ( f `  x )  =  1o ) )
8584, 49eleq2s 2288 . . . . . . . 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 799 . . . . . 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 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 } ) )
8988ssrdv 3185 . . . 4  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  C  C_ 
{ A ,  B } )
90 prssi 3776 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
9125, 12, 90syl2anc 411 . . . 4  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  { A ,  B }  C_  C
)
9289, 91eqssd 3196 . . 3  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  C  =  { A ,  B } )
934, 92exlimddv 1910 . 2  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  =  { A ,  B }
)
9493ex 115 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 104    <-> wb 105    \/ wo 709    /\ w3a 980    = wceq 1364   E.wex 1503    e. wcel 2164    =/= wne 2364    C_ wss 3153   (/)c0 3446   {cpr 3619   class class class wbr 4029   -->wf 5250   -1-1->wf1 5251   -1-1-onto->wf1o 5253   ` cfv 5254   1oc1o 6462   2oc2o 6463    ~~ cen 6792
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1o 6469  df-2o 6470  df-en 6795
This theorem is referenced by:  exmidpw  6964  en2eleq  7255  isprm2lem  12254
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