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Theorem hashtpg 12636
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 1014 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  C  e.  _V )
2 prfi 7833 . . . . . . 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 3952 . . . . . . . . . . . . . . . 16  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  <->  ( C  =  A  \/  C  =  B ) ) )
5 orcom 393 . . . . . . . . . . . . . . . . 17  |-  ( ( C  =  A  \/  C  =  B )  <->  ( C  =  B  \/  C  =  A )
)
6 nne 2628 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  B  =/=  C  <->  B  =  C )
7 eqcom 2459 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  C  <->  C  =  B )
86, 7bitri 257 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  B  =/=  C  <->  C  =  B )
98bicomi 207 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  B  <->  -.  B  =/=  C )
10 nne 2628 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  C  =/=  A  <->  C  =  A )
1110bicomi 207 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  A  <->  -.  C  =/=  A )
129, 11orbi12i 528 . . . . . . . . . . . . . . . . 17  |-  ( ( C  =  B  \/  C  =  A )  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) )
135, 12bitri 257 . . . . . . . . . . . . . . . 16  |-  ( ( C  =  A  \/  C  =  B )  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) )
144, 13syl6bb 269 . . . . . . . . . . . . . . 15  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  <->  ( -.  B  =/=  C  \/  -.  C  =/=  A ) ) )
1514biimpd 212 . . . . . . . . . . . . . 14  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  ->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
16153ad2ant3 1032 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
1716imp 435 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  C  e.  { A ,  B } )  -> 
( -.  B  =/= 
C  \/  -.  C  =/=  A ) )
1817olcd 399 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  C  e.  { A ,  B } )  -> 
( -.  A  =/= 
B  \/  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
1918ex 440 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  A  =/=  B  \/  ( -.  B  =/= 
C  \/  -.  C  =/=  A ) ) ) )
20 3orass 989 . . . . . . . . . 10  |-  ( ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
)  <->  ( -.  A  =/=  B  \/  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
2119, 20syl6ibr 235 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
22 3ianor 1003 . . . . . . . . 9  |-  ( -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  <->  ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A ) )
2321, 22syl6ibr 235 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  -.  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) ) )
2423con2d 120 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  -.  C  e.  { A ,  B } ) )
2524imp 435 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  -.  C  e.  { A ,  B } )
26 hashunsng 12565 . . . . . . 7  |-  ( C  e.  _V  ->  (
( { A ,  B }  e.  Fin  /\ 
-.  C  e.  { A ,  B }
)  ->  ( # `  ( { A ,  B }  u.  { C } ) )  =  ( (
# `  { A ,  B } )  +  1 ) ) )
2726imp 435 . . . . . 6  |-  ( ( C  e.  _V  /\  ( { A ,  B }  e.  Fin  /\  -.  C  e.  { A ,  B } ) )  ->  ( # `  ( { A ,  B }  u.  { C } ) )  =  ( (
# `  { A ,  B } )  +  1 ) )
281, 3, 25, 27syl12anc 1269 . . . . 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 1015 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  A  =/=  B )
30 3simpa 1006 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( A  e.  _V  /\  B  e.  _V ) )
3130adantr 471 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( A  e.  _V  /\  B  e.  _V ) )
32 hashprg 12566 . . . . . . . 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 215 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 { A ,  B } )  =  2 )
3534oveq1d 6291 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  (
( # `  { A ,  B } )  +  1 )  =  ( 2  +  1 ) )
3628, 35eqtrd 2486 . . . 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 3941 . . . . 5  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
3837fveq2i 5851 . . . 4  |-  ( # `  { A ,  B ,  C } )  =  ( # `  ( { A ,  B }  u.  { C } ) )
39 df-3 10658 . . . 4  |-  3  =  ( 2  +  1 )
4036, 38, 393eqtr4g 2511 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 { A ,  B ,  C }
)  =  3 )
4140ex 440 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( # `  { A ,  B ,  C } )  =  3 ) )
42 nne 2628 . . . . . . 7  |-  ( -.  A  =/=  B  <->  A  =  B )
43 hashprlei 12624 . . . . . . . . . 10  |-  ( { B ,  C }  e.  Fin  /\  ( # `  { B ,  C } )  <_  2
)
44 prfi 7833 . . . . . . . . . . . . . . 15  |-  { B ,  C }  e.  Fin
45 hashcl 12532 . . . . . . . . . . . . . . . 16  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  NN0 )
4645nn0zd 11028 . . . . . . . . . . . . . . 15  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  ZZ )
4744, 46ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { B ,  C } )  e.  ZZ
48 2z 10959 . . . . . . . . . . . . . 14  |-  2  e.  ZZ
49 zleltp1 10977 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <_  2  <->  ( # `  { B ,  C }
)  <  ( 2  +  1 ) ) )
50 2p1e3 10723 . . . . . . . . . . . . . . . . . 18  |-  ( 2  +  1 )  =  3
5150a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( 2  +  1 )  =  3 )
5251breq2d 4386 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <  ( 2  +  1 )  <->  ( # `  { B ,  C }
)  <  3 ) )
5352biimpd 212 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <  ( 2  +  1 )  -> 
( # `  { B ,  C } )  <  3 ) )
5449, 53sylbid 223 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <_  2  ->  (
# `  { B ,  C } )  <  3 ) )
5547, 48, 54mp2an 683 . . . . . . . . . . . . 13  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  ( # `  { B ,  C }
)  <  3 )
5645nn0red 10916 . . . . . . . . . . . . . . 15  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  RR )
5744, 56ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { B ,  C } )  e.  RR
58 3re 10672 . . . . . . . . . . . . . 14  |-  3  e.  RR
5957, 58ltnei 9745 . . . . . . . . . . . . 13  |-  ( (
# `  { B ,  C } )  <  3  ->  3  =/=  ( # `  { B ,  C } ) )
6055, 59syl 17 . . . . . . . . . . . 12  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  3  =/=  ( # `  { B ,  C } ) )
6160necomd 2679 . . . . . . . . . . 11  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  ( # `  { B ,  C }
)  =/=  3 )
6261adantl 472 . . . . . . . . . 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 4029 . . . . . . . . . . 11  |-  ( A  =  B  ->  { A ,  B ,  C }  =  { B ,  B ,  C } )
66 tpidm12 4042 . . . . . . . . . . 11  |-  { B ,  B ,  C }  =  { B ,  C }
6765, 66syl6req 2503 . . . . . . . . . 10  |-  ( A  =  B  ->  { B ,  C }  =  { A ,  B ,  C } )
6867fveq2d 5852 . . . . . . . . 9  |-  ( A  =  B  ->  ( # `
 { B ,  C } )  =  (
# `  { A ,  B ,  C }
) )
6968neeq1d 2683 . . . . . . . 8  |-  ( A  =  B  ->  (
( # `  { B ,  C } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
7064, 69syl5ib 227 . . . . . . 7  |-  ( A  =  B  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
7142, 70sylbi 200 . . . . . 6  |-  ( -.  A  =/=  B  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
72 hashprlei 12624 . . . . . . . . . 10  |-  ( { A ,  C }  e.  Fin  /\  ( # `  { A ,  C } )  <_  2
)
73 prfi 7833 . . . . . . . . . . . . . . 15  |-  { A ,  C }  e.  Fin
74 hashcl 12532 . . . . . . . . . . . . . . . 16  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  NN0 )
7574nn0zd 11028 . . . . . . . . . . . . . . 15  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  ZZ )
7673, 75ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  C } )  e.  ZZ
77 zleltp1 10977 . . . . . . . . . . . . . . 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 4386 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <  ( 2  +  1 )  <->  ( # `  { A ,  C }
)  <  3 ) )
8079biimpd 212 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <  ( 2  +  1 )  -> 
( # `  { A ,  C } )  <  3 ) )
8177, 80sylbid 223 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { A ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  C }
)  <_  2  ->  (
# `  { A ,  C } )  <  3 ) )
8276, 48, 81mp2an 683 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  ( # `  { A ,  C }
)  <  3 )
8374nn0red 10916 . . . . . . . . . . . . . . 15  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  RR )
8473, 83ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  C } )  e.  RR
8584, 58ltnei 9745 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  C } )  <  3  ->  3  =/=  ( # `  { A ,  C } ) )
8682, 85syl 17 . . . . . . . . . . . 12  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  3  =/=  ( # `  { A ,  C } ) )
8786necomd 2679 . . . . . . . . . . 11  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  ( # `  { A ,  C }
)  =/=  3 )
8887adantl 472 . . . . . . . . . 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 4030 . . . . . . . . . . 11  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  C ,  C } )
92 tpidm23 4044 . . . . . . . . . . 11  |-  { A ,  C ,  C }  =  { A ,  C }
9391, 92syl6req 2503 . . . . . . . . . 10  |-  ( B  =  C  ->  { A ,  C }  =  { A ,  B ,  C } )
9493fveq2d 5852 . . . . . . . . 9  |-  ( B  =  C  ->  ( # `
 { A ,  C } )  =  (
# `  { A ,  B ,  C }
) )
9594neeq1d 2683 . . . . . . . 8  |-  ( B  =  C  ->  (
( # `  { A ,  C } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
9690, 95syl5ib 227 . . . . . . 7  |-  ( B  =  C  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
976, 96sylbi 200 . . . . . 6  |-  ( -.  B  =/=  C  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
98 hashprlei 12624 . . . . . . . . . 10  |-  ( { A ,  B }  e.  Fin  /\  ( # `  { A ,  B } )  <_  2
)
99 hashcl 12532 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  NN0 )
10099nn0zd 11028 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  ZZ )
1012, 100ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  B } )  e.  ZZ
102 zleltp1 10977 . . . . . . . . . . . . . . 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 4386 . . . . . . . . . . . . . . . 16  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <  ( 2  +  1 )  <->  ( # `  { A ,  B }
)  <  3 ) )
105104biimpd 212 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <  ( 2  +  1 )  -> 
( # `  { A ,  B } )  <  3 ) )
106102, 105sylbid 223 . . . . . . . . . . . . . 14  |-  ( ( ( # `  { A ,  B }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { A ,  B }
)  <_  2  ->  (
# `  { A ,  B } )  <  3 ) )
107101, 48, 106mp2an 683 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  ( # `  { A ,  B }
)  <  3 )
10899nn0red 10916 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  RR )
1092, 108ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  B } )  e.  RR
110109, 58ltnei 9745 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  B } )  <  3  ->  3  =/=  ( # `  { A ,  B } ) )
111107, 110syl 17 . . . . . . . . . . . 12  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  3  =/=  ( # `  { A ,  B } ) )
112111necomd 2679 . . . . . . . . . . 11  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  ( # `  { A ,  B }
)  =/=  3 )
113112adantl 472 . . . . . . . . . 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 4031 . . . . . . . . . . 11  |-  ( C  =  A  ->  { A ,  B ,  C }  =  { A ,  B ,  A } )
117 tpidm13 4043 . . . . . . . . . . 11  |-  { A ,  B ,  A }  =  { A ,  B }
118116, 117syl6req 2503 . . . . . . . . . 10  |-  ( C  =  A  ->  { A ,  B }  =  { A ,  B ,  C } )
119118fveq2d 5852 . . . . . . . . 9  |-  ( C  =  A  ->  ( # `
 { A ,  B } )  =  (
# `  { A ,  B ,  C }
) )
120119neeq1d 2683 . . . . . . . 8  |-  ( C  =  A  ->  (
( # `  { A ,  B } )  =/=  3  <->  ( # `  { A ,  B ,  C } )  =/=  3
) )
121115, 120syl5ib 227 . . . . . . 7  |-  ( C  =  A  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
12210, 121sylbi 200 . . . . . 6  |-  ( -.  C  =/=  A  -> 
( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( # `  { A ,  B ,  C } )  =/=  3
) )
12371, 97, 1223jaoi 1335 . . . . 5  |-  ( ( -.  A  =/=  B  \/  -.  B  =/=  C  \/  -.  C  =/=  A
)  ->  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e. 
_V )  ->  ( # `
 { A ,  B ,  C }
)  =/=  3 ) )
12422, 123sylbi 200 . . . 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 2644 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( # `  { A ,  B ,  C }
)  =  3  -> 
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) ) )
12741, 126impbid 195 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 189    \/ wo 374    /\ wa 375    \/ w3o 985    /\ w3a 986    = wceq 1448    e. wcel 1891    =/= wne 2622   _Vcvv 3013    u. cun 3370   {csn 3936   {cpr 3938   {ctp 3940   class class class wbr 4374   ` cfv 5561  (class class class)co 6276   Fincfn 7556   RRcr 9525   1c1 9527    + caddc 9529    < clt 9662    <_ cle 9663   2c2 10648   3c3 10649   ZZcz 10927   #chash 12509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571  ax-cnex 9582  ax-resscn 9583  ax-1cn 9584  ax-icn 9585  ax-addcl 9586  ax-addrcl 9587  ax-mulcl 9588  ax-mulrcl 9589  ax-mulcom 9590  ax-addass 9591  ax-mulass 9592  ax-distr 9593  ax-i2m1 9594  ax-1ne0 9595  ax-1rid 9596  ax-rnegex 9597  ax-rrecex 9598  ax-cnre 9599  ax-pre-lttri 9600  ax-pre-lttrn 9601  ax-pre-ltadd 9602  ax-pre-mulgt0 9603
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-int 4205  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-pred 5359  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-om 6681  df-1st 6781  df-2nd 6782  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-card 8360  df-cda 8585  df-pnf 9664  df-mnf 9665  df-xr 9666  df-ltxr 9667  df-le 9668  df-sub 9849  df-neg 9850  df-nn 10599  df-2 10657  df-3 10658  df-n0 10860  df-z 10928  df-uz 11150  df-fz 11776  df-hash 12510
This theorem is referenced by:  hashge3el3dif  12637  constr3lem2  25386  poimirlem9  31951
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