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Theorem hashtpg 12510
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 999 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  C  e.  _V )
2 prfi 7787 . . . . . . 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 4032 . . . . . . . . . . . . . . . 16  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  <->  ( C  =  A  \/  C  =  B ) ) )
5 orcom 385 . . . . . . . . . . . . . . . . 17  |-  ( ( C  =  A  \/  C  =  B )  <->  ( C  =  B  \/  C  =  A )
)
6 nne 2655 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  B  =/=  C  <->  B  =  C )
7 eqcom 2463 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  C  <->  C  =  B )
86, 7bitri 249 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  B  =/=  C  <->  C  =  B )
98bicomi 202 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  B  <->  -.  B  =/=  C )
10 nne 2655 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  C  =/=  A  <->  C  =  A )
1110bicomi 202 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  A  <->  -.  C  =/=  A )
129, 11orbi12i 519 . . . . . . . . . . . . . . . . 17  |-  ( ( C  =  B  \/  C  =  A )  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) )
135, 12bitri 249 . . . . . . . . . . . . . . . 16  |-  ( ( C  =  A  \/  C  =  B )  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) )
144, 13syl6bb 261 . . . . . . . . . . . . . . 15  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A ) ) )
1514biimpd 207 . . . . . . . . . . . . . 14  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  ->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
16153ad2ant3 1017 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
1716imp 427 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  C  e.  { A ,  B } )  -> 
( -.  B  =/= 
C  \/  -.  C  =/=  A ) )
1817olcd 391 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  C  e.  { A ,  B } )  -> 
( -.  A  =/= 
B  \/  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
1918ex 432 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  A  =/=  B  \/  ( -.  B  =/= 
C  \/  -.  C  =/=  A ) ) ) )
20 3orass 974 . . . . . . . . . 10  |-  ( ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
)  <->  ( -.  A  =/=  B  \/  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
2119, 20syl6ibr 227 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
22 3ianor 988 . . . . . . . . 9  |-  ( -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  <->  ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A ) )
2321, 22syl6ibr 227 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) ) )
2423con2d 115 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  -.  C  e.  { A ,  B } ) )
2524imp 427 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  -.  C  e.  { A ,  B } )
26 hashunsng 12446 . . . . . . 7  |-  ( C  e.  _V  ->  (
( { A ,  B }  e.  Fin  /\ 
-.  C  e.  { A ,  B }
)  ->  ( # `  ( { A ,  B }  u.  { C } ) )  =  ( (
# `  { A ,  B } )  +  1 ) ) )
2726imp 427 . . . . . 6  |-  ( ( C  e.  _V  /\  ( { A ,  B }  e.  Fin  /\  -.  C  e.  { A ,  B } ) )  ->  ( # `  ( { A ,  B }  u.  { C } ) )  =  ( (
# `  { A ,  B } )  +  1 ) )
281, 3, 25, 27syl12anc 1224 . . . . 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 1000 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  A  =/=  B )
30 3simpa 991 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( A  e.  _V  /\  B  e.  _V ) )
3130adantr 463 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( A  e.  _V  /\  B  e.  _V ) )
32 hashprg 12447 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A  =/=  B  <->  (
# `  { A ,  B } )  =  2 ) )
3331, 32syl 16 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( A  =/=  B  <->  ( # `  { A ,  B }
)  =  2 ) )
3429, 33mpbid 210 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 { A ,  B } )  =  2 )
3534oveq1d 6285 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  (
( # `  { A ,  B } )  +  1 )  =  ( 2  +  1 ) )
3628, 35eqtrd 2495 . . . 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 4021 . . . . 5  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
3837fveq2i 5851 . . . 4  |-  ( # `  { A ,  B ,  C } )  =  ( # `  ( { A ,  B }  u.  { C } ) )
39 df-3 10591 . . . 4  |-  3  =  ( 2  +  1 )
4036, 38, 393eqtr4g 2520 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 { A ,  B ,  C }
)  =  3 )
4140ex 432 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( # `  { A ,  B ,  C } )  =  3 ) )
42 nne 2655 . . . . . . 7  |-  ( -.  A  =/=  B  <->  A  =  B )
43 hashprlei 12501 . . . . . . . . . 10  |-  ( { B ,  C }  e.  Fin  /\  ( # `  { B ,  C } )  <_  2
)
44 prfi 7787 . . . . . . . . . . . . . . 15  |-  { B ,  C }  e.  Fin
45 hashcl 12413 . . . . . . . . . . . . . . . 16  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  NN0 )
4645nn0zd 10963 . . . . . . . . . . . . . . 15  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  ZZ )
4744, 46ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { B ,  C } )  e.  ZZ
48 2z 10892 . . . . . . . . . . . . . 14  |-  2  e.  ZZ
49 zleltp1 10910 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <_  2  <->  ( # `  { B ,  C }
)  <  ( 2  +  1 ) ) )
50 2p1e3 10655 . . . . . . . . . . . . . . . . . 18  |-  ( 2  +  1 )  =  3
5150a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( 2  +  1 )  =  3 )
5251breq2d 4451 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <  ( 2  +  1 )  <->  ( # `  { B ,  C }
)  <  3 ) )
5352biimpd 207 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <  ( 2  +  1 )  -> 
( # `  { B ,  C } )  <  3 ) )
5449, 53sylbid 215 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <_  2  ->  (
# `  { B ,  C } )  <  3 ) )
5547, 48, 54mp2an 670 . . . . . . . . . . . . 13  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  ( # `  { B ,  C }
)  <  3 )
5645nn0red 10849 . . . . . . . . . . . . . . 15  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  RR )
5744, 56ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { B ,  C } )  e.  RR
58 3re 10605 . . . . . . . . . . . . . 14  |-  3  e.  RR
5957, 58ltnei 9697 . . . . . . . . . . . . 13  |-  ( (
# `  { B ,  C } )  <  3  ->  3  =/=  ( # `  { B ,  C } ) )
6055, 59syl 16 . . . . . . . . . . . 12  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  3  =/=  ( # `  { B ,  C } ) )
6160necomd 2725 . . . . . . . . . . 11  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  ( # `  { B ,  C }
)  =/=  3 )
6261adantl 464 . . . . . . . . . 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 4104 . . . . . . . . . . 11  |-  ( A  =  B  ->  { A ,  B ,  C }  =  { B ,  B ,  C } )
66 tpidm12 4117 . . . . . . . . . . 11  |-  { B ,  B ,  C }  =  { B ,  C }
6765, 66syl6req 2512 . . . . . . . . . 10  |-  ( A  =  B  ->  { B ,  C }  =  { A ,  B ,  C } )
6867fveq2d 5852 . . . . . . . . 9  |-  ( A  =  B  ->  ( # `
 { B ,  C } )  =  (
# `  { A ,  B ,  C }
) )
6968neeq1d 2731 . . . . . . . 8  |-  ( A  =  B  ->  (
( # `  { B ,  C } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
7064, 69syl5ib 219 . . . . . . 7  |-  ( A  =  B  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
7142, 70sylbi 195 . . . . . 6  |-  ( -.  A  =/=  B  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
72 hashprlei 12501 . . . . . . . . . 10  |-  ( { A ,  C }  e.  Fin  /\  ( # `  { A ,  C } )  <_  2
)
73 prfi 7787 . . . . . . . . . . . . . . 15  |-  { A ,  C }  e.  Fin
74 hashcl 12413 . . . . . . . . . . . . . . . 16  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  NN0 )
7574nn0zd 10963 . . . . . . . . . . . . . . 15  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  ZZ )
7673, 75ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  C } )  e.  ZZ
77 zleltp1 10910 . . . . . . . . . . . . . . 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 4451 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <  ( 2  +  1 )  <->  ( # `  { A ,  C }
)  <  3 ) )
8079biimpd 207 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <  ( 2  +  1 )  -> 
( # `  { A ,  C } )  <  3 ) )
8177, 80sylbid 215 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <_  2  ->  (
# `  { A ,  C } )  <  3 ) )
8276, 48, 81mp2an 670 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  ( # `  { A ,  C }
)  <  3 )
8374nn0red 10849 . . . . . . . . . . . . . . 15  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  RR )
8473, 83ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  C } )  e.  RR
8584, 58ltnei 9697 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  C } )  <  3  ->  3  =/=  ( # `  { A ,  C } ) )
8682, 85syl 16 . . . . . . . . . . . 12  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  3  =/=  ( # `  { A ,  C } ) )
8786necomd 2725 . . . . . . . . . . 11  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  ( # `  { A ,  C }
)  =/=  3 )
8887adantl 464 . . . . . . . . . 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 4105 . . . . . . . . . . 11  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  C ,  C } )
92 tpidm23 4119 . . . . . . . . . . 11  |-  { A ,  C ,  C }  =  { A ,  C }
9391, 92syl6req 2512 . . . . . . . . . 10  |-  ( B  =  C  ->  { A ,  C }  =  { A ,  B ,  C } )
9493fveq2d 5852 . . . . . . . . 9  |-  ( B  =  C  ->  ( # `
 { A ,  C } )  =  (
# `  { A ,  B ,  C }
) )
9594neeq1d 2731 . . . . . . . 8  |-  ( B  =  C  ->  (
( # `  { A ,  C } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
9690, 95syl5ib 219 . . . . . . 7  |-  ( B  =  C  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
976, 96sylbi 195 . . . . . 6  |-  ( -.  B  =/=  C  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
98 hashprlei 12501 . . . . . . . . . 10  |-  ( { A ,  B }  e.  Fin  /\  ( # `  { A ,  B } )  <_  2
)
99 hashcl 12413 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  NN0 )
10099nn0zd 10963 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  ZZ )
1012, 100ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  B } )  e.  ZZ
102 zleltp1 10910 . . . . . . . . . . . . . . 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 4451 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <  ( 2  +  1 )  <->  ( # `  { A ,  B }
)  <  3 ) )
105104biimpd 207 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <  ( 2  +  1 )  -> 
( # `  { A ,  B } )  <  3 ) )
106102, 105sylbid 215 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <_  2  ->  (
# `  { A ,  B } )  <  3 ) )
107101, 48, 106mp2an 670 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  ( # `  { A ,  B }
)  <  3 )
10899nn0red 10849 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  RR )
1092, 108ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  B } )  e.  RR
110109, 58ltnei 9697 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  B } )  <  3  ->  3  =/=  ( # `  { A ,  B } ) )
111107, 110syl 16 . . . . . . . . . . . 12  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  3  =/=  ( # `  { A ,  B } ) )
112111necomd 2725 . . . . . . . . . . 11  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  ( # `  { A ,  B }
)  =/=  3 )
113112adantl 464 . . . . . . . . . 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 4106 . . . . . . . . . . 11  |-  ( C  =  A  ->  { A ,  B ,  C }  =  { A ,  B ,  A } )
117 tpidm13 4118 . . . . . . . . . . 11  |-  { A ,  B ,  A }  =  { A ,  B }
118116, 117syl6req 2512 . . . . . . . . . 10  |-  ( C  =  A  ->  { A ,  B }  =  { A ,  B ,  C } )
119118fveq2d 5852 . . . . . . . . 9  |-  ( C  =  A  ->  ( # `
 { A ,  B } )  =  (
# `  { A ,  B ,  C }
) )
120119neeq1d 2731 . . . . . . . 8  |-  ( C  =  A  ->  (
( # `  { A ,  B } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
121115, 120syl5ib 219 . . . . . . 7  |-  ( C  =  A  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
12210, 121sylbi 195 . . . . . 6  |-  ( -.  C  =/=  A  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
12371, 97, 1223jaoi 1289 . . . . 5  |-  ( ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V )  ->  ( # `
 { A ,  B ,  C }
)  =/=  3 ) )
12422, 123sylbi 195 . . . 4  |-  ( -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V )  ->  ( # `
 { A ,  B ,  C }
)  =/=  3 ) )
125124com12 31 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
126125necon4bd 2676 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( # `  { A ,  B ,  C }
)  =  3  -> 
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) ) )
12741, 126impbid 191 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 184    \/ wo 366    /\ wa 367    \/ w3o 970    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    u. cun 3459   {csn 4016   {cpr 4018   {ctp 4020   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Fincfn 7509   RRcr 9480   1c1 9482    + caddc 9484    < clt 9617    <_ cle 9618   2c2 10581   3c3 10582   ZZcz 10860   #chash 12390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12391
This theorem is referenced by:  hashge3el3dif  12511  constr3lem2  24851
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