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Theorem hashtpg 12632
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 simpl3 1010 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  C  e.  _V )
2 prfi 7852 . . . . . . 7  |-  { A ,  B }  e.  Fin
32a1i 11 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  { A ,  B }  e.  Fin )
4 elprg 4018 . . . . . . . . . . . . . . . 16  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  <->  ( C  =  A  \/  C  =  B ) ) )
5 orcom 388 . . . . . . . . . . . . . . . . 17  |-  ( ( C  =  A  \/  C  =  B )  <->  ( C  =  B  \/  C  =  A )
)
6 nne 2631 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  B  =/=  C  <->  B  =  C )
7 eqcom 2438 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  C  <->  C  =  B )
86, 7bitri 252 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  B  =/=  C  <->  C  =  B )
98bicomi 205 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  B  <->  -.  B  =/=  C )
10 nne 2631 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  C  =/=  A  <->  C  =  A )
1110bicomi 205 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  A  <->  -.  C  =/=  A )
129, 11orbi12i 523 . . . . . . . . . . . . . . . . 17  |-  ( ( C  =  B  \/  C  =  A )  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) )
135, 12bitri 252 . . . . . . . . . . . . . . . 16  |-  ( ( C  =  A  \/  C  =  B )  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) )
144, 13syl6bb 264 . . . . . . . . . . . . . . 15  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A ) ) )
1514biimpd 210 . . . . . . . . . . . . . 14  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  ->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
16153ad2ant3 1028 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
1716imp 430 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  C  e.  { A ,  B } )  -> 
( -.  B  =/= 
C  \/  -.  C  =/=  A ) )
1817olcd 394 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  C  e.  { A ,  B } )  -> 
( -.  A  =/= 
B  \/  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
1918ex 435 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  A  =/=  B  \/  ( -.  B  =/= 
C  \/  -.  C  =/=  A ) ) ) )
20 3orass 985 . . . . . . . . . 10  |-  ( ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
)  <->  ( -.  A  =/=  B  \/  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
2119, 20syl6ibr 230 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
22 3ianor 999 . . . . . . . . 9  |-  ( -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  <->  ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A ) )
2321, 22syl6ibr 230 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) ) )
2423con2d 118 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  -.  C  e.  { A ,  B } ) )
2524imp 430 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  -.  C  e.  { A ,  B } )
26 hashunsng 12568 . . . . . . 7  |-  ( C  e.  _V  ->  (
( { A ,  B }  e.  Fin  /\ 
-.  C  e.  { A ,  B }
)  ->  ( # `  ( { A ,  B }  u.  { C } ) )  =  ( (
# `  { A ,  B } )  +  1 ) ) )
2726imp 430 . . . . . 6  |-  ( ( C  e.  _V  /\  ( { A ,  B }  e.  Fin  /\  -.  C  e.  { A ,  B } ) )  ->  ( # `  ( { A ,  B }  u.  { C } ) )  =  ( (
# `  { A ,  B } )  +  1 ) )
281, 3, 25, 27syl12anc 1262 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 ( { A ,  B }  u.  { C } ) )  =  ( ( # `  { A ,  B }
)  +  1 ) )
29 simpr1 1011 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  A  =/=  B )
30 3simpa 1002 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( A  e.  _V  /\  B  e.  _V ) )
3130adantr 466 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( A  e.  _V  /\  B  e.  _V ) )
32 hashprg 12569 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  =/=  B  <->  (
# `  { A ,  B } )  =  2 ) )
3331, 32syl 17 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( A  =/=  B  <->  ( # `  { A ,  B }
)  =  2 ) )
3429, 33mpbid 213 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 { A ,  B } )  =  2 )
3534oveq1d 6320 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  (
( # `  { A ,  B } )  +  1 )  =  ( 2  +  1 ) )
3628, 35eqtrd 2470 . . . 4  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 ( { A ,  B }  u.  { C } ) )  =  ( 2  +  1 ) )
37 df-tp 4007 . . . . 5  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
3837fveq2i 5884 . . . 4  |-  ( # `  { A ,  B ,  C } )  =  ( # `  ( { A ,  B }  u.  { C } ) )
39 df-3 10669 . . . 4  |-  3  =  ( 2  +  1 )
4036, 38, 393eqtr4g 2495 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 { A ,  B ,  C }
)  =  3 )
4140ex 435 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( # `  { A ,  B ,  C } )  =  3 ) )
42 nne 2631 . . . . . . 7  |-  ( -.  A  =/=  B  <->  A  =  B )
43 hashprlei 12623 . . . . . . . . . 10  |-  ( { B ,  C }  e.  Fin  /\  ( # `  { B ,  C } )  <_  2
)
44 prfi 7852 . . . . . . . . . . . . . . 15  |-  { B ,  C }  e.  Fin
45 hashcl 12535 . . . . . . . . . . . . . . . 16  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  NN0 )
4645nn0zd 11038 . . . . . . . . . . . . . . 15  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  ZZ )
4744, 46ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { B ,  C } )  e.  ZZ
48 2z 10969 . . . . . . . . . . . . . 14  |-  2  e.  ZZ
49 zleltp1 10987 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <_  2  <->  ( # `  { B ,  C }
)  <  ( 2  +  1 ) ) )
50 2p1e3 10733 . . . . . . . . . . . . . . . . . 18  |-  ( 2  +  1 )  =  3
5150a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( 2  +  1 )  =  3 )
5251breq2d 4438 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <  ( 2  +  1 )  <->  ( # `  { B ,  C }
)  <  3 ) )
5352biimpd 210 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <  ( 2  +  1 )  -> 
( # `  { B ,  C } )  <  3 ) )
5449, 53sylbid 218 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <_  2  ->  (
# `  { B ,  C } )  <  3 ) )
5547, 48, 54mp2an 676 . . . . . . . . . . . . 13  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  ( # `  { B ,  C }
)  <  3 )
5645nn0red 10926 . . . . . . . . . . . . . . 15  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  RR )
5744, 56ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { B ,  C } )  e.  RR
58 3re 10683 . . . . . . . . . . . . . 14  |-  3  e.  RR
5957, 58ltnei 9757 . . . . . . . . . . . . 13  |-  ( (
# `  { B ,  C } )  <  3  ->  3  =/=  ( # `  { B ,  C } ) )
6055, 59syl 17 . . . . . . . . . . . 12  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  3  =/=  ( # `  { B ,  C } ) )
6160necomd 2702 . . . . . . . . . . 11  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  ( # `  { B ,  C }
)  =/=  3 )
6261adantl 467 . . . . . . . . . 10  |-  ( ( { B ,  C }  e.  Fin  /\  ( # `
 { B ,  C } )  <_  2
)  ->  ( # `  { B ,  C }
)  =/=  3 )
6343, 62ax-mp 5 . . . . . . . . 9  |-  ( # `  { B ,  C } )  =/=  3
6463a1i 11 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `
 { B ,  C } )  =/=  3
)
65 tpeq1 4091 . . . . . . . . . . 11  |-  ( A  =  B  ->  { A ,  B ,  C }  =  { B ,  B ,  C } )
66 tpidm12 4104 . . . . . . . . . . 11  |-  { B ,  B ,  C }  =  { B ,  C }
6765, 66syl6req 2487 . . . . . . . . . 10  |-  ( A  =  B  ->  { B ,  C }  =  { A ,  B ,  C } )
6867fveq2d 5885 . . . . . . . . 9  |-  ( A  =  B  ->  ( # `
 { B ,  C } )  =  (
# `  { A ,  B ,  C }
) )
6968neeq1d 2708 . . . . . . . 8  |-  ( A  =  B  ->  (
( # `  { B ,  C } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
7064, 69syl5ib 222 . . . . . . 7  |-  ( A  =  B  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
7142, 70sylbi 198 . . . . . 6  |-  ( -.  A  =/=  B  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
72 hashprlei 12623 . . . . . . . . . 10  |-  ( { A ,  C }  e.  Fin  /\  ( # `  { A ,  C } )  <_  2
)
73 prfi 7852 . . . . . . . . . . . . . . 15  |-  { A ,  C }  e.  Fin
74 hashcl 12535 . . . . . . . . . . . . . . . 16  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  NN0 )
7574nn0zd 11038 . . . . . . . . . . . . . . 15  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  ZZ )
7673, 75ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  C } )  e.  ZZ
77 zleltp1 10987 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <_  2  <->  ( # `  { A ,  C }
)  <  ( 2  +  1 ) ) )
7850a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( 2  +  1 )  =  3 )
7978breq2d 4438 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <  ( 2  +  1 )  <->  ( # `  { A ,  C }
)  <  3 ) )
8079biimpd 210 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <  ( 2  +  1 )  -> 
( # `  { A ,  C } )  <  3 ) )
8177, 80sylbid 218 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <_  2  ->  (
# `  { A ,  C } )  <  3 ) )
8276, 48, 81mp2an 676 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  ( # `  { A ,  C }
)  <  3 )
8374nn0red 10926 . . . . . . . . . . . . . . 15  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  RR )
8473, 83ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  C } )  e.  RR
8584, 58ltnei 9757 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  C } )  <  3  ->  3  =/=  ( # `  { A ,  C } ) )
8682, 85syl 17 . . . . . . . . . . . 12  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  3  =/=  ( # `  { A ,  C } ) )
8786necomd 2702 . . . . . . . . . . 11  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  ( # `  { A ,  C }
)  =/=  3 )
8887adantl 467 . . . . . . . . . 10  |-  ( ( { A ,  C }  e.  Fin  /\  ( # `
 { A ,  C } )  <_  2
)  ->  ( # `  { A ,  C }
)  =/=  3 )
8972, 88ax-mp 5 . . . . . . . . 9  |-  ( # `  { A ,  C } )  =/=  3
9089a1i 11 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `
 { A ,  C } )  =/=  3
)
91 tpeq2 4092 . . . . . . . . . . 11  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  C ,  C } )
92 tpidm23 4106 . . . . . . . . . . 11  |-  { A ,  C ,  C }  =  { A ,  C }
9391, 92syl6req 2487 . . . . . . . . . 10  |-  ( B  =  C  ->  { A ,  C }  =  { A ,  B ,  C } )
9493fveq2d 5885 . . . . . . . . 9  |-  ( B  =  C  ->  ( # `
 { A ,  C } )  =  (
# `  { A ,  B ,  C }
) )
9594neeq1d 2708 . . . . . . . 8  |-  ( B  =  C  ->  (
( # `  { A ,  C } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
9690, 95syl5ib 222 . . . . . . 7  |-  ( B  =  C  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
976, 96sylbi 198 . . . . . 6  |-  ( -.  B  =/=  C  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
98 hashprlei 12623 . . . . . . . . . 10  |-  ( { A ,  B }  e.  Fin  /\  ( # `  { A ,  B } )  <_  2
)
99 hashcl 12535 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  NN0 )
10099nn0zd 11038 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  ZZ )
1012, 100ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  B } )  e.  ZZ
102 zleltp1 10987 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <_  2  <->  ( # `  { A ,  B }
)  <  ( 2  +  1 ) ) )
10350a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( 2  +  1 )  =  3 )
104103breq2d 4438 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <  ( 2  +  1 )  <->  ( # `  { A ,  B }
)  <  3 ) )
105104biimpd 210 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <  ( 2  +  1 )  -> 
( # `  { A ,  B } )  <  3 ) )
106102, 105sylbid 218 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <_  2  ->  (
# `  { A ,  B } )  <  3 ) )
107101, 48, 106mp2an 676 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  ( # `  { A ,  B }
)  <  3 )
10899nn0red 10926 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  RR )
1092, 108ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  B } )  e.  RR
110109, 58ltnei 9757 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  B } )  <  3  ->  3  =/=  ( # `  { A ,  B } ) )
111107, 110syl 17 . . . . . . . . . . . 12  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  3  =/=  ( # `  { A ,  B } ) )
112111necomd 2702 . . . . . . . . . . 11  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  ( # `  { A ,  B }
)  =/=  3 )
113112adantl 467 . . . . . . . . . 10  |-  ( ( { A ,  B }  e.  Fin  /\  ( # `
 { A ,  B } )  <_  2
)  ->  ( # `  { A ,  B }
)  =/=  3 )
11498, 113ax-mp 5 . . . . . . . . 9  |-  ( # `  { A ,  B } )  =/=  3
115114a1i 11 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `
 { A ,  B } )  =/=  3
)
116 tpeq3 4093 . . . . . . . . . . 11  |-  ( C  =  A  ->  { A ,  B ,  C }  =  { A ,  B ,  A } )
117 tpidm13 4105 . . . . . . . . . . 11  |-  { A ,  B ,  A }  =  { A ,  B }
118116, 117syl6req 2487 . . . . . . . . . 10  |-  ( C  =  A  ->  { A ,  B }  =  { A ,  B ,  C } )
119118fveq2d 5885 . . . . . . . . 9  |-  ( C  =  A  ->  ( # `
 { A ,  B } )  =  (
# `  { A ,  B ,  C }
) )
120119neeq1d 2708 . . . . . . . 8  |-  ( C  =  A  ->  (
( # `  { A ,  B } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
121115, 120syl5ib 222 . . . . . . 7  |-  ( C  =  A  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
12210, 121sylbi 198 . . . . . 6  |-  ( -.  C  =/=  A  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
12371, 97, 1223jaoi 1327 . . . . 5  |-  ( ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V )  ->  ( # `
 { A ,  B ,  C }
)  =/=  3 ) )
12422, 123sylbi 198 . . . 4  |-  ( -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V )  ->  ( # `
 { A ,  B ,  C }
)  =/=  3 ) )
125124com12 32 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
126125necon4bd 2653 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( # `  { A ,  B ,  C }
)  =  3  -> 
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) ) )
12741, 126impbid 193 1  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  <->  ( # `  { A ,  B ,  C } )  =  3 ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   _Vcvv 3087    u. cun 3440   {csn 4002   {cpr 4004   {ctp 4006   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Fincfn 7577   RRcr 9537   1c1 9539    + caddc 9541    < clt 9674    <_ cle 9675   2c2 10659   3c3 10660   ZZcz 10937   #chash 12512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-hash 12513
This theorem is referenced by:  hashge3el3dif  12633  constr3lem2  25219  poimirlem9  31652
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