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Theorem en2eqpr 6808
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 6648 . . . . . 6  |-  ( C 
~~  2o  <->  E. f  f : C -1-1-onto-> 2o )
21biimpi 119 . . . . 5  |-  ( C 
~~  2o  ->  E. f 
f : C -1-1-onto-> 2o )
323ad2ant1 1003 . . . 4  |-  ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  ->  E. f  f : C -1-1-onto-> 2o )
43adantr 274 . . 3  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  E. f 
f : C -1-1-onto-> 2o )
5 simplr 520 . . . . . . . . . . 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 )  =  (/) )
75, 6eqtr4d 2176 . . . . . . . . . 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 5373 . . . . . . . . . . . . . 14  |-  ( f : C -1-1-onto-> 2o  ->  f : C -1-1-> 2o )
98adantl 275 . . . . . . . . . . . . 13  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  f : C -1-1-> 2o )
109adantr 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 1023 . . . . . . . . . . . . 13  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  B  e.  C )
1312adantr 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 5680 . . . . . . . . . . . 12  |-  ( ( f : C -1-1-> 2o  /\  ( x  e.  C  /\  B  e.  C
) )  ->  (
( f `  x
)  =  ( f `
 B )  <->  x  =  B ) )
1510, 11, 13, 14syl12anc 1215 . . . . . . . . . . 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 480 . . . . . . . . . 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 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 3635 . . . . . . . . . . 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 480 . . . . . . . . 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 2217 . . . . . . . 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 524 . . . . . . . . . . . 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 )  =  (/) )
2422, 23eqtr4d 2176 . . . . . . . . . . 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 1022 . . . . . . . . . . . . . 14  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  A  e.  C )
2625adantr 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 5680 . . . . . . . . . . . . 13  |-  ( ( f : C -1-1-> 2o  /\  ( x  e.  C  /\  A  e.  C
) )  ->  (
( f `  x
)  =  ( f `
 A )  <->  x  =  A ) )
2810, 11, 26, 27syl12anc 1215 . . . . . . . . . . . 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 484 . . . . . . . . . . 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 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 3634 . . . . . . . . . . . 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 484 . . . . . . . . . 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 2217 . . . . . . . . 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 520 . . . . . . . . . . 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 2176 . . . . . . . . . 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 520 . . . . . . . . . . . . 13  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  A  =/=  B )
3938neneqd 2330 . . . . . . . . . . . 12  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  -.  A  =  B )
40 f1fveq 5680 . . . . . . . . . . . . 13  |-  ( ( f : C -1-1-> 2o  /\  ( A  e.  C  /\  B  e.  C
) )  ->  (
( f `  A
)  =  ( f `
 B )  <->  A  =  B ) )
419, 25, 12, 40syl12anc 1215 . . . . . . . . . . . 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 663 . . . . . . . . . . 11  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  -.  ( f `  A
)  =  ( f `
 B ) )
4342ad4antr 486 . . . . . . . . . 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 610 . . . . . . . . 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 5374 . . . . . . . . . . . . 13  |-  ( f : C -1-1-onto-> 2o  ->  f : C
--> 2o )
4645adantl 275 . . . . . . . . . . . 12  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  f : C --> 2o )
4746, 25ffvelrnd 5563 . . . . . . . . . . 11  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
f `  A )  e.  2o )
48 elpri 3554 . . . . . . . . . . . 12  |-  ( ( f `  A )  e.  { (/) ,  1o }  ->  ( ( f `
 A )  =  (/)  \/  ( f `  A )  =  1o ) )
49 df2o3 6334 . . . . . . . . . . . 12  |-  2o  =  { (/) ,  1o }
5048, 49eleq2s 2235 . . . . . . . . . . 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 484 . . . . . . . . 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 788 . . . . . . . 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 5563 . . . . . . . . . 10  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
f `  B )  e.  2o )
55 elpri 3554 . . . . . . . . . . 11  |-  ( ( f `  B )  e.  { (/) ,  1o }  ->  ( ( f `
 B )  =  (/)  \/  ( f `  B )  =  1o ) )
5655, 49eleq2s 2235 . . . . . . . . . 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 480 . . . . . . . 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 788 . . . . . . 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 520 . . . . . . . . . . 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 2176 . . . . . . . . . 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 486 . . . . . . . . . 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 610 . . . . . . . . 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 524 . . . . . . . . . . . 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 )
6765, 66eqtr4d 2176 . . . . . . . . . . 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 484 . . . . . . . . . . 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 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 )
7032ad3antrrr 484 . . . . . . . . . 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 2217 . . . . . . . . 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 484 . . . . . . . . 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 788 . . . . . . . 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 520 . . . . . . . . . . 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 )
7674, 75eqtr4d 2176 . . . . . . . . . 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 480 . . . . . . . . . 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 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 )
7919ad2antrr 480 . . . . . . . . 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 2217 . . . . . . . 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 480 . . . . . . . 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 788 . . . . . . 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 5562 . . . . . . . 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 3554 . . . . . . . . 9  |-  ( ( f `  x )  e.  { (/) ,  1o }  ->  ( ( f `
 x )  =  (/)  \/  ( f `  x )  =  1o ) )
8584, 49eleq2s 2235 . . . . . . . 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 788 . . . . . 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 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 } ) )
8988ssrdv 3107 . . . 4  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  C  C_ 
{ A ,  B } )
90 prssi 3685 . . . . 5  |-  ( ( A  e.  C  /\  B  e.  C )  ->  { A ,  B }  C_  C )
9125, 12, 90syl2anc 409 . . . 4  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  { A ,  B }  C_  C
)
9289, 91eqssd 3118 . . 3  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  C  =  { A ,  B } )
934, 92exlimddv 1871 . 2  |-  ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C )  /\  A  =/=  B
)  ->  C  =  { A ,  B }
)
9493ex 114 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 103    <-> wb 104    \/ wo 698    /\ w3a 963    = wceq 1332   E.wex 1469    e. wcel 1481    =/= wne 2309    C_ wss 3075   (/)c0 3367   {cpr 3532   class class class wbr 3936   -->wf 5126   -1-1->wf1 5127   -1-1-onto->wf1o 5129   ` cfv 5130   1oc1o 6313   2oc2o 6314    ~~ cen 6639
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-suc 4300  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-1o 6320  df-2o 6321  df-en 6642
This theorem is referenced by:  exmidpw  6809  en2eleq  7067  isprm2lem  11831
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