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Theorem en2eqpr 6881
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 6721 . . . . . 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 1013 . . . 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 525 . . . . . . . . . . 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 2206 . . . . . . . . . 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 5439 . . . . . . . . . . . . . 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 1033 . . . . . . . . . . . . 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 5748 . . . . . . . . . . . 12  |-  ( ( f : C -1-1-> 2o  /\  ( x  e.  C  /\  B  e.  C
) )  ->  (
( f `  x
)  =  ( f `
 B )  <->  x  =  B ) )
1510, 11, 13, 14syl12anc 1231 . . . . . . . . . . 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 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 )  =  (/) )  ->  (
( 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 3686 . . . . . . . . . . 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 485 . . . . . . . . 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 2247 . . . . . . . 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 529 . . . . . . . . . . . 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 2206 . . . . . . . . . . 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 1032 . . . . . . . . . . . . . 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 5748 . . . . . . . . . . . . 13  |-  ( ( f : C -1-1-> 2o  /\  ( x  e.  C  /\  A  e.  C
) )  ->  (
( f `  x
)  =  ( f `
 A )  <->  x  =  A ) )
2810, 11, 26, 27syl12anc 1231 . . . . . . . . . . . 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 489 . . . . . . . . . . 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 3685 . . . . . . . . . . . 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 489 . . . . . . . . . 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 2247 . . . . . . . . 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 525 . . . . . . . . . . 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 2206 . . . . . . . . . 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 525 . . . . . . . . . . . . 13  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  A  =/=  B )
3938neneqd 2361 . . . . . . . . . . . 12  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  -.  A  =  B )
40 f1fveq 5748 . . . . . . . . . . . . 13  |-  ( ( f : C -1-1-> 2o  /\  ( A  e.  C  /\  B  e.  C
) )  ->  (
( f `  A
)  =  ( f `
 B )  <->  A  =  B ) )
419, 25, 12, 40syl12anc 1231 . . . . . . . . . . . 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 668 . . . . . . . . . . 11  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  -.  ( f `  A
)  =  ( f `
 B ) )
4342ad4antr 491 . . . . . . . . . 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 615 . . . . . . . . 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 5440 . . . . . . . . . . . . 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 5629 . . . . . . . . . . 11  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
f `  A )  e.  2o )
48 elpri 3604 . . . . . . . . . . . 12  |-  ( ( f `  A )  e.  { (/) ,  1o }  ->  ( ( f `
 A )  =  (/)  \/  ( f `  A )  =  1o ) )
49 df2o3 6406 . . . . . . . . . . . 12  |-  2o  =  { (/) ,  1o }
5048, 49eleq2s 2265 . . . . . . . . . . 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 489 . . . . . . . . 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 793 . . . . . . . 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 5629 . . . . . . . . . 10  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  (
f `  B )  e.  2o )
55 elpri 3604 . . . . . . . . . . 11  |-  ( ( f `  B )  e.  { (/) ,  1o }  ->  ( ( f `
 B )  =  (/)  \/  ( f `  B )  =  1o ) )
5655, 49eleq2s 2265 . . . . . . . . . 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 485 . . . . . . . 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 793 . . . . . . 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 525 . . . . . . . . . . 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 2206 . . . . . . . . . 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 491 . . . . . . . . . 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 615 . . . . . . . . 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 529 . . . . . . . . . . . 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 2206 . . . . . . . . . . 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 489 . . . . . . . . . . 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 489 . . . . . . . . . 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 2247 . . . . . . . . 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 489 . . . . . . . . 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 793 . . . . . . . 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 525 . . . . . . . . . . 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 2206 . . . . . . . . . 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 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 )  =  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 485 . . . . . . . . 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 2247 . . . . . . . 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 485 . . . . . . . 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 793 . . . . . . 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 5628 . . . . . . . 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 3604 . . . . . . . . 9  |-  ( ( f `  x )  e.  { (/) ,  1o }  ->  ( ( f `
 x )  =  (/)  \/  ( f `  x )  =  1o ) )
8584, 49eleq2s 2265 . . . . . . . 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 793 . . . . . 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 3153 . . . 4  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  C  C_ 
{ A ,  B } )
90 prssi 3736 . . . . 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 3164 . . 3  |-  ( ( ( ( C  ~~  2o  /\  A  e.  C  /\  B  e.  C
)  /\  A  =/=  B )  /\  f : C -1-1-onto-> 2o )  ->  C  =  { A ,  B } )
934, 92exlimddv 1891 . 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 703    /\ w3a 973    = wceq 1348   E.wex 1485    e. wcel 2141    =/= wne 2340    C_ wss 3121   (/)c0 3414   {cpr 3582   class class class wbr 3987   -->wf 5192   -1-1->wf1 5193   -1-1-onto->wf1o 5195   ` cfv 5196   1oc1o 6385   2oc2o 6386    ~~ cen 6712
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-suc 4354  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1o 6392  df-2o 6393  df-en 6715
This theorem is referenced by:  exmidpw  6882  en2eleq  7159  isprm2lem  12057
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