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Theorem hashtpg 12191
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 993 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  C  e.  _V )
2 prfi 7591 . . . . . . 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 3898 . . . . . . . . . . . . . . . 16  |-  ( C  e.  _V  ->  ( C  e.  { A ,  B }  <->  ( C  =  A  \/  C  =  B ) ) )
5 orcom 387 . . . . . . . . . . . . . . . . 17  |-  ( ( C  =  A  \/  C  =  B )  <->  ( C  =  B  \/  C  =  A )
)
6 nne 2617 . . . . . . . . . . . . . . . . . . . 20  |-  ( -.  B  =/=  C  <->  B  =  C )
7 eqcom 2445 . . . . . . . . . . . . . . . . . . . 20  |-  ( B  =  C  <->  C  =  B )
86, 7bitri 249 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  B  =/=  C  <->  C  =  B )
98bicomi 202 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  B  <->  -.  B  =/=  C )
10 nne 2617 . . . . . . . . . . . . . . . . . . 19  |-  ( -.  C  =/=  A  <->  C  =  A )
1110bicomi 202 . . . . . . . . . . . . . . . . . 18  |-  ( C  =  A  <->  -.  C  =/=  A )
129, 11orbi12i 521 . . . . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . 13  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
1716imp 429 . . . . . . . . . . . 12  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  C  e.  { A ,  B } )  -> 
( -.  B  =/= 
C  \/  -.  C  =/=  A ) )
1817olcd 393 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  C  e.  { A ,  B } )  -> 
( -.  A  =/= 
B  \/  ( -.  B  =/=  C  \/  -.  C  =/=  A
) ) )
1918ex 434 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  { A ,  B }  ->  ( -.  A  =/=  B  \/  ( -.  B  =/= 
C  \/  -.  C  =/=  A ) ) ) )
20 3orass 968 . . . . . . . . . 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 982 . . . . . . . . 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 429 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  -.  C  e.  { A ,  B } )
26 hashunsng 12159 . . . . . . 7  |-  ( C  e.  _V  ->  (
( { A ,  B }  e.  Fin  /\ 
-.  C  e.  { A ,  B }
)  ->  ( # `  ( { A ,  B }  u.  { C } ) )  =  ( (
# `  { A ,  B } )  +  1 ) ) )
2726imp 429 . . . . . 6  |-  ( ( C  e.  _V  /\  ( { A ,  B }  e.  Fin  /\  -.  C  e.  { A ,  B } ) )  ->  ( # `  ( { A ,  B }  u.  { C } ) )  =  ( (
# `  { A ,  B } )  +  1 ) )
281, 3, 25, 27syl12anc 1216 . . . . 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 994 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  A  =/=  B )
30 3simpa 985 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( A  e.  _V  /\  B  e.  _V ) )
3130adantr 465 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( A  e.  _V  /\  B  e.  _V ) )
32 hashprg 12160 . . . . . . . 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 6111 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  (
( # `  { A ,  B } )  +  1 )  =  ( 2  +  1 ) )
3628, 35eqtrd 2475 . . . 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 3887 . . . . 5  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
3837fveq2i 5699 . . . 4  |-  ( # `  { A ,  B ,  C } )  =  ( # `  ( { A ,  B }  u.  { C } ) )
39 df-3 10386 . . . 4  |-  3  =  ( 2  +  1 )
4036, 38, 393eqtr4g 2500 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
) )  ->  ( # `
 { A ,  B ,  C }
)  =  3 )
4140ex 434 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  (
( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A
)  ->  ( # `  { A ,  B ,  C } )  =  3 ) )
42 nne 2617 . . . . . . 7  |-  ( -.  A  =/=  B  <->  A  =  B )
43 hashprlei 12182 . . . . . . . . . 10  |-  ( { B ,  C }  e.  Fin  /\  ( # `  { B ,  C } )  <_  2
)
44 prfi 7591 . . . . . . . . . . . . . . 15  |-  { B ,  C }  e.  Fin
45 hashcl 12131 . . . . . . . . . . . . . . . 16  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  NN0 )
4645nn0zd 10750 . . . . . . . . . . . . . . 15  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  ZZ )
4744, 46ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { B ,  C } )  e.  ZZ
48 2z 10683 . . . . . . . . . . . . . 14  |-  2  e.  ZZ
49 zleltp1 10700 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( ( # `  { B ,  C }
)  <_  2  <->  ( # `  { B ,  C }
)  <  ( 2  +  1 ) ) )
50 2p1e3 10450 . . . . . . . . . . . . . . . . . 18  |-  ( 2  +  1 )  =  3
5150a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( # `  { B ,  C }
)  e.  ZZ  /\  2  e.  ZZ )  ->  ( 2  +  1 )  =  3 )
5251breq2d 4309 . . . . . . . . . . . . . . . 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 672 . . . . . . . . . . . . 13  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  ( # `  { B ,  C }
)  <  3 )
5645nn0red 10642 . . . . . . . . . . . . . . 15  |-  ( { B ,  C }  e.  Fin  ->  ( # `  { B ,  C }
)  e.  RR )
5744, 56ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { B ,  C } )  e.  RR
58 3re 10400 . . . . . . . . . . . . . 14  |-  3  e.  RR
5957, 58ltnei 9503 . . . . . . . . . . . . 13  |-  ( (
# `  { B ,  C } )  <  3  ->  3  =/=  ( # `  { B ,  C } ) )
6055, 59syl 16 . . . . . . . . . . . 12  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  3  =/=  ( # `  { B ,  C } ) )
6160necomd 2700 . . . . . . . . . . 11  |-  ( (
# `  { B ,  C } )  <_ 
2  ->  ( # `  { B ,  C }
)  =/=  3 )
6261adantl 466 . . . . . . . . . 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 3968 . . . . . . . . . . 11  |-  ( A  =  B  ->  { A ,  B ,  C }  =  { B ,  B ,  C } )
66 tpidm12 3981 . . . . . . . . . . 11  |-  { B ,  B ,  C }  =  { B ,  C }
6765, 66syl6req 2492 . . . . . . . . . 10  |-  ( A  =  B  ->  { B ,  C }  =  { A ,  B ,  C } )
6867fveq2d 5700 . . . . . . . . 9  |-  ( A  =  B  ->  ( # `
 { B ,  C } )  =  (
# `  { A ,  B ,  C }
) )
6968neeq1d 2626 . . . . . . . 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 12182 . . . . . . . . . 10  |-  ( { A ,  C }  e.  Fin  /\  ( # `  { A ,  C } )  <_  2
)
73 prfi 7591 . . . . . . . . . . . . . . 15  |-  { A ,  C }  e.  Fin
74 hashcl 12131 . . . . . . . . . . . . . . . 16  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  NN0 )
7574nn0zd 10750 . . . . . . . . . . . . . . 15  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  ZZ )
7673, 75ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  C } )  e.  ZZ
77 zleltp1 10700 . . . . . . . . . . . . . . 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 4309 . . . . . . . . . . . . . . . 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 672 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  ( # `  { A ,  C }
)  <  3 )
8374nn0red 10642 . . . . . . . . . . . . . . 15  |-  ( { A ,  C }  e.  Fin  ->  ( # `  { A ,  C }
)  e.  RR )
8473, 83ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  C } )  e.  RR
8584, 58ltnei 9503 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  C } )  <  3  ->  3  =/=  ( # `  { A ,  C } ) )
8682, 85syl 16 . . . . . . . . . . . 12  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  3  =/=  ( # `  { A ,  C } ) )
8786necomd 2700 . . . . . . . . . . 11  |-  ( (
# `  { A ,  C } )  <_ 
2  ->  ( # `  { A ,  C }
)  =/=  3 )
8887adantl 466 . . . . . . . . . 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 3969 . . . . . . . . . . 11  |-  ( B  =  C  ->  { A ,  B ,  C }  =  { A ,  C ,  C } )
92 tpidm23 3983 . . . . . . . . . . 11  |-  { A ,  C ,  C }  =  { A ,  C }
9391, 92syl6req 2492 . . . . . . . . . 10  |-  ( B  =  C  ->  { A ,  C }  =  { A ,  B ,  C } )
9493fveq2d 5700 . . . . . . . . 9  |-  ( B  =  C  ->  ( # `
 { A ,  C } )  =  (
# `  { A ,  B ,  C }
) )
9594neeq1d 2626 . . . . . . . 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 12182 . . . . . . . . . 10  |-  ( { A ,  B }  e.  Fin  /\  ( # `  { A ,  B } )  <_  2
)
99 hashcl 12131 . . . . . . . . . . . . . . . 16  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  NN0 )
10099nn0zd 10750 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  ZZ )
1012, 100ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  B } )  e.  ZZ
102 zleltp1 10700 . . . . . . . . . . . . . . 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 4309 . . . . . . . . . . . . . . . 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 672 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  ( # `  { A ,  B }
)  <  3 )
10899nn0red 10642 . . . . . . . . . . . . . . 15  |-  ( { A ,  B }  e.  Fin  ->  ( # `  { A ,  B }
)  e.  RR )
1092, 108ax-mp 5 . . . . . . . . . . . . . 14  |-  ( # `  { A ,  B } )  e.  RR
110109, 58ltnei 9503 . . . . . . . . . . . . 13  |-  ( (
# `  { A ,  B } )  <  3  ->  3  =/=  ( # `  { A ,  B } ) )
111107, 110syl 16 . . . . . . . . . . . 12  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  3  =/=  ( # `  { A ,  B } ) )
112111necomd 2700 . . . . . . . . . . 11  |-  ( (
# `  { A ,  B } )  <_ 
2  ->  ( # `  { A ,  B }
)  =/=  3 )
113112adantl 466 . . . . . . . . . 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 3970 . . . . . . . . . . 11  |-  ( C  =  A  ->  { A ,  B ,  C }  =  { A ,  B ,  A } )
117 tpidm13 3982 . . . . . . . . . . 11  |-  { A ,  B ,  A }  =  { A ,  B }
118116, 117syl6req 2492 . . . . . . . . . 10  |-  ( C  =  A  ->  { A ,  B }  =  { A ,  B ,  C } )
119118fveq2d 5700 . . . . . . . . 9  |-  ( C  =  A  ->  ( # `
 { A ,  B } )  =  (
# `  { A ,  B ,  C }
) )
120119neeq1d 2626 . . . . . . . 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 1281 . . . . 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 2678 . 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 368    /\ wa 369    \/ w3o 964    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   _Vcvv 2977    u. cun 3331   {csn 3882   {cpr 3884   {ctp 3886   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Fincfn 7315   RRcr 9286   1c1 9288    + caddc 9290    < clt 9423    <_ cle 9424   2c2 10376   3c3 10377   ZZcz 10651   #chash 12108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-card 8114  df-cda 8342  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-hash 12109
This theorem is referenced by:  hashge3el3dif  12192  constr3lem2  23537
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