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Theorem hashtpg 27279
Description: The size of an unordered triple of three different elements. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
Assertion
Ref Expression
hashtpg  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  <->  ( # `  { A ,  B ,  C } )  =  3 ) )

Proof of Theorem hashtpg
StepHypRef Expression
1 id 19 . . . . . . . 8  |-  ( C  e.  _V  ->  C  e.  _V )
213ad2ant3 978 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  C  e.  _V )
32adantr 451 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  C  e.  _V )
4 prfi 7176 . . . . . . 7  |-  { A ,  B }  e.  Fin
54a1i 10 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  { A ,  B }  e.  Fin )
6 elprg 3691 . . . . . . . . . . . . . . . 16  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  <->  ( C  =  A  \/  C  =  B ) ) )
7 orcom 376 . . . . . . . . . . . . . . . . 17  |-  ( ( C  =  A  \/  C  =  B )  <->  ( C  =  B  \/  C  =  A )
)
8 nne 2483 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  B  =/=  C  <->  B  =  C )
9 eqcom 2318 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  C  <->  C  =  B )
108, 9bitri 240 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  B  =/=  C  <->  C  =  B )
1110bicomi 193 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  B  <->  -.  B  =/=  C )
12 nne 2483 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  C  =/=  A  <->  C  =  A )
1312bicomi 193 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  A  <->  -.  C  =/=  A )
1411, 13orbi12i 507 . . . . . . . . . . . . . . . . 17  |-  ( ( C  =  B  \/  C  =  A )  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) )
157, 14bitri 240 . . . . . . . . . . . . . . . 16  |-  ( ( C  =  A  \/  C  =  B )  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) )
166, 15syl6bb 252 . . . . . . . . . . . . . . 15  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A ) ) )
1716biimpd 198 . . . . . . . . . . . . . 14  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  ->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
18173ad2ant3 978 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
1918imp 418 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  C  e.  { A ,  B } )  -> 
( -.  B  =/= 
C  \/  -.  C  =/=  A ) )
2019olcd 382 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  C  e.  { A ,  B } )  -> 
( -.  A  =/= 
B  \/  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
2120ex 423 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  A  =/=  B  \/  ( -.  B  =/= 
C  \/  -.  C  =/=  A ) ) ) )
22 3orass 937 . . . . . . . . . 10  |-  ( ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
)  <->  ( -.  A  =/=  B  \/  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
2321, 22syl6ibr 218 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
24 3ianor 949 . . . . . . . . 9  |-  ( -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  <->  ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A ) )
2523, 24syl6ibr 218 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) ) )
2625con2d 107 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  -.  C  e.  { A ,  B } ) )
2726imp 418 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  -.  C  e.  { A ,  B } )
28 hashunsng 11414 . . . . . . 7  |-  ( C  e.  _V  ->  (
( { A ,  B }  e.  Fin  /\ 
-.  C  e.  { A ,  B }
)  ->  ( # `  ( { A ,  B }  u.  { C } ) )  =  ( (
# `  { A ,  B } )  +  1 ) ) )
2928imp 418 . . . . . 6  |-  ( ( C  e.  _V  /\  ( { A ,  B }  e.  Fin  /\  -.  C  e.  { A ,  B } ) )  ->  ( # `  ( { A ,  B }  u.  { C } ) )  =  ( (
# `  { A ,  B } )  +  1 ) )
303, 5, 27, 29syl12anc 1180 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 ( { A ,  B }  u.  { C } ) )  =  ( ( # `  { A ,  B }
)  +  1 ) )
31 simpr1 961 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  A  =/=  B )
32 3simpa 952 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( A  e.  _V  /\  B  e.  _V ) )
3332adantr 451 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( A  e.  _V  /\  B  e.  _V ) )
34 hashprg 11415 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  =/=  B  <->  (
# `  { A ,  B } )  =  2 ) )
3533, 34syl 15 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( A  =/=  B  <->  ( # `  { A ,  B }
)  =  2 ) )
3631, 35mpbid 201 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 { A ,  B } )  =  2 )
3736oveq1d 5915 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  (
( # `  { A ,  B } )  +  1 )  =  ( 2  +  1 ) )
3830, 37eqtrd 2348 . . . 4  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 ( { A ,  B }  u.  { C } ) )  =  ( 2  +  1 ) )
39 df-tp 3682 . . . . 5  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
4039fveq2i 5566 . . . 4  |-  ( # `  { A ,  B ,  C } )  =  ( # `  ( { A ,  B }  u.  { C } ) )
41 df-3 9850 . . . 4  |-  3  =  ( 2  +  1 )
4238, 40, 413eqtr4g 2373 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 { A ,  B ,  C }
)  =  3 )
4342ex 423 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( # `  { A ,  B ,  C } )  =  3 ) )
44 nne 2483 . . . . . . 7  |-  ( -.  A  =/=  B  <->  A  =  B )
45 hashprlei 11426 . . . . . . . . . 10  |-  ( { B ,  C }  e.  Fin  /\  ( # `  { B ,  C } )  <_  2
)
46 prfi 7176 . . . . . . . . . . . . . . 15  |-  { B ,  C }  e.  Fin
47 hashcl 11397 . . . . . . . . . . . . . . . 16  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  NN0 )
4847nn0zd 10162 . . . . . . . . . . . . . . 15  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  ZZ )
4946, 48ax-mp 8 . . . . . . . . . . . . . 14  |-  ( # `  { B ,  C } )  e.  ZZ
50 2z 10101 . . . . . . . . . . . . . 14  |-  2  e.  ZZ
51 zleltp1 10115 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <_  2  <->  ( # `  { B ,  C }
)  <  ( 2  +  1 ) ) )
52 2p1e3 9894 . . . . . . . . . . . . . . . . . 18  |-  ( 2  +  1 )  =  3
5352a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( 2  +  1 )  =  3 )
5453breq2d 4072 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <  ( 2  +  1 )  <->  ( # `  { B ,  C }
)  <  3 ) )
5554biimpd 198 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <  ( 2  +  1 )  -> 
( # `  { B ,  C } )  <  3 ) )
5651, 55sylbid 206 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <_  2  ->  (
# `  { B ,  C } )  <  3 ) )
5749, 50, 56mp2an 653 . . . . . . . . . . . . 13  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  ( # `  { B ,  C }
)  <  3 )
5847nn0red 10066 . . . . . . . . . . . . . . 15  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  RR )
5946, 58ax-mp 8 . . . . . . . . . . . . . 14  |-  ( # `  { B ,  C } )  e.  RR
60 3re 9862 . . . . . . . . . . . . . 14  |-  3  e.  RR
6159, 60ltnei 8988 . . . . . . . . . . . . 13  |-  ( (
# `  { B ,  C } )  <  3  ->  3  =/=  ( # `  { B ,  C } ) )
6257, 61syl 15 . . . . . . . . . . . 12  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  3  =/=  ( # `  { B ,  C } ) )
6362necomd 2562 . . . . . . . . . . 11  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  ( # `  { B ,  C }
)  =/=  3 )
6463adantl 452 . . . . . . . . . 10  |-  ( ( { B ,  C }  e.  Fin  /\  ( # `
 { B ,  C } )  <_  2
)  ->  ( # `  { B ,  C }
)  =/=  3 )
6545, 64ax-mp 8 . . . . . . . . 9  |-  ( # `  { B ,  C } )  =/=  3
6665a1i 10 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `
 { B ,  C } )  =/=  3
)
67 tpeq1 3749 . . . . . . . . . . 11  |-  ( A  =  B  ->  { A ,  B ,  C }  =  { B ,  B ,  C } )
68 tpidm12 3762 . . . . . . . . . . 11  |-  { B ,  B ,  C }  =  { B ,  C }
6967, 68syl6req 2365 . . . . . . . . . 10  |-  ( A  =  B  ->  { B ,  C }  =  { A ,  B ,  C } )
7069fveq2d 5567 . . . . . . . . 9  |-  ( A  =  B  ->  ( # `
 { B ,  C } )  =  (
# `  { A ,  B ,  C }
) )
7170neeq1d 2492 . . . . . . . 8  |-  ( A  =  B  ->  (
( # `  { B ,  C } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
7266, 71syl5ib 210 . . . . . . 7  |-  ( A  =  B  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
7344, 72sylbi 187 . . . . . 6  |-  ( -.  A  =/=  B  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
74 hashprlei 11426 . . . . . . . . . 10  |-  ( { A ,  C }  e.  Fin  /\  ( # `  { A ,  C } )  <_  2
)
75 prfi 7176 . . . . . . . . . . . . . . 15  |-  { A ,  C }  e.  Fin
76 hashcl 11397 . . . . . . . . . . . . . . . 16  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  NN0 )
7776nn0zd 10162 . . . . . . . . . . . . . . 15  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  ZZ )
7875, 77ax-mp 8 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  C } )  e.  ZZ
79 zleltp1 10115 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <_  2  <->  ( # `  { A ,  C }
)  <  ( 2  +  1 ) ) )
8052a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( 2  +  1 )  =  3 )
8180breq2d 4072 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <  ( 2  +  1 )  <->  ( # `  { A ,  C }
)  <  3 ) )
8281biimpd 198 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <  ( 2  +  1 )  -> 
( # `  { A ,  C } )  <  3 ) )
8379, 82sylbid 206 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <_  2  ->  (
# `  { A ,  C } )  <  3 ) )
8478, 50, 83mp2an 653 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  ( # `  { A ,  C }
)  <  3 )
8576nn0red 10066 . . . . . . . . . . . . . . 15  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  RR )
8675, 85ax-mp 8 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  C } )  e.  RR
8786, 60ltnei 8988 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  C } )  <  3  ->  3  =/=  ( # `  { A ,  C } ) )
8884, 87syl 15 . . . . . . . . . . . 12  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  3  =/=  ( # `  { A ,  C } ) )
8988necomd 2562 . . . . . . . . . . 11  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  ( # `  { A ,  C }
)  =/=  3 )
9089adantl 452 . . . . . . . . . 10  |-  ( ( { A ,  C }  e.  Fin  /\  ( # `
 { A ,  C } )  <_  2
)  ->  ( # `  { A ,  C }
)  =/=  3 )
9174, 90ax-mp 8 . . . . . . . . 9  |-  ( # `  { A ,  C } )  =/=  3
9291a1i 10 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `
 { A ,  C } )  =/=  3
)
93 tpeq2 3750 . . . . . . . . . . 11  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  C ,  C } )
94 tpidm23 3764 . . . . . . . . . . 11  |-  { A ,  C ,  C }  =  { A ,  C }
9593, 94syl6req 2365 . . . . . . . . . 10  |-  ( B  =  C  ->  { A ,  C }  =  { A ,  B ,  C } )
9695fveq2d 5567 . . . . . . . . 9  |-  ( B  =  C  ->  ( # `
 { A ,  C } )  =  (
# `  { A ,  B ,  C }
) )
9796neeq1d 2492 . . . . . . . 8  |-  ( B  =  C  ->  (
( # `  { A ,  C } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
9892, 97syl5ib 210 . . . . . . 7  |-  ( B  =  C  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
998, 98sylbi 187 . . . . . 6  |-  ( -.  B  =/=  C  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
100 hashprlei 11426 . . . . . . . . . 10  |-  ( { A ,  B }  e.  Fin  /\  ( # `  { A ,  B } )  <_  2
)
101 hashcl 11397 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  NN0 )
102101nn0zd 10162 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  ZZ )
1034, 102ax-mp 8 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  B } )  e.  ZZ
104 zleltp1 10115 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <_  2  <->  ( # `  { A ,  B }
)  <  ( 2  +  1 ) ) )
10552a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( 2  +  1 )  =  3 )
106105breq2d 4072 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <  ( 2  +  1 )  <->  ( # `  { A ,  B }
)  <  3 ) )
107106biimpd 198 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <  ( 2  +  1 )  -> 
( # `  { A ,  B } )  <  3 ) )
108104, 107sylbid 206 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <_  2  ->  (
# `  { A ,  B } )  <  3 ) )
109103, 50, 108mp2an 653 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  ( # `  { A ,  B }
)  <  3 )
110101nn0red 10066 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  RR )
1114, 110ax-mp 8 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  B } )  e.  RR
112111, 60ltnei 8988 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  B } )  <  3  ->  3  =/=  ( # `  { A ,  B } ) )
113109, 112syl 15 . . . . . . . . . . . 12  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  3  =/=  ( # `  { A ,  B } ) )
114113necomd 2562 . . . . . . . . . . 11  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  ( # `  { A ,  B }
)  =/=  3 )
115114adantl 452 . . . . . . . . . 10  |-  ( ( { A ,  B }  e.  Fin  /\  ( # `
 { A ,  B } )  <_  2
)  ->  ( # `  { A ,  B }
)  =/=  3 )
116100, 115ax-mp 8 . . . . . . . . 9  |-  ( # `  { A ,  B } )  =/=  3
117116a1i 10 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `
 { A ,  B } )  =/=  3
)
118 tpeq3 3751 . . . . . . . . . . 11  |-  ( C  =  A  ->  { A ,  B ,  C }  =  { A ,  B ,  A } )
119 tpidm13 3763 . . . . . . . . . . 11  |-  { A ,  B ,  A }  =  { A ,  B }
120118, 119syl6req 2365 . . . . . . . . . 10  |-  ( C  =  A  ->  { A ,  B }  =  { A ,  B ,  C } )
121120fveq2d 5567 . . . . . . . . 9  |-  ( C  =  A  ->  ( # `
 { A ,  B } )  =  (
# `  { A ,  B ,  C }
) )
122121neeq1d 2492 . . . . . . . 8  |-  ( C  =  A  ->  (
( # `  { A ,  B } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
123117, 122syl5ib 210 . . . . . . 7  |-  ( C  =  A  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
12412, 123sylbi 187 . . . . . 6  |-  ( -.  C  =/=  A  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
12573, 99, 1243jaoi 1245 . . . . 5  |-  ( ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V )  ->  ( # `
 { A ,  B ,  C }
)  =/=  3 ) )
12624, 125sylbi 187 . . . 4  |-  ( -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V )  ->  ( # `
 { A ,  B ,  C }
)  =/=  3 ) )
127126com12 27 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
128127necon4bd 2541 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( # `  { A ,  B ,  C }
)  =  3  -> 
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) ) )
12943, 128impbid 183 1  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  <->  ( # `  { A ,  B ,  C } )  =  3 ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   _Vcvv 2822    u. cun 3184   {csn 3674   {cpr 3675   {ctp 3676   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Fincfn 6906   RRcr 8781   1c1 8783    + caddc 8785    < clt 8912    <_ cle 8913   2c2 9840   3c3 9841   ZZcz 10071   #chash 11384
This theorem is referenced by:  constr3lem2  27530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-hash 11385
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