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Theorem hashdifpr 10559
Description: The size of the difference of a finite set and a proper ordered pair subset is the set's size minus 2. (Contributed by AV, 16-Dec-2020.)
Assertion
Ref Expression
hashdifpr  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  ( `  ( A  \  { B ,  C }
) )  =  ( ( `  A )  -  2 ) )

Proof of Theorem hashdifpr
StepHypRef Expression
1 difpr 3657 . . . 4  |-  ( A 
\  { B ,  C } )  =  ( ( A  \  { B } )  \  { C } )
21a1i 9 . . 3  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  ( A  \  { B ,  C } )  =  ( ( A  \  { B } )  \  { C } ) )
32fveq2d 5418 . 2  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  ( `  ( A  \  { B ,  C }
) )  =  ( `  ( ( A  \  { B } )  \  { C } ) ) )
4 simpl 108 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  A  e.  Fin )
5 snfig 6701 . . . . . 6  |-  ( B  e.  A  ->  { B }  e.  Fin )
653ad2ant1 1002 . . . . 5  |-  ( ( B  e.  A  /\  C  e.  A  /\  B  =/=  C )  ->  { B }  e.  Fin )
76adantl 275 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  { B }  e.  Fin )
8 snssi 3659 . . . . . 6  |-  ( B  e.  A  ->  { B }  C_  A )
983ad2ant1 1002 . . . . 5  |-  ( ( B  e.  A  /\  C  e.  A  /\  B  =/=  C )  ->  { B }  C_  A
)
109adantl 275 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  { B }  C_  A )
11 diffifi 6781 . . . 4  |-  ( ( A  e.  Fin  /\  { B }  e.  Fin  /\ 
{ B }  C_  A )  ->  ( A  \  { B }
)  e.  Fin )
124, 7, 10, 11syl3anc 1216 . . 3  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  ( A  \  { B }
)  e.  Fin )
13 simpr2 988 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  C  e.  A )
14 simpr3 989 . . . . 5  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  B  =/=  C )
1514necomd 2392 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  C  =/=  B )
16 eldifsn 3645 . . . 4  |-  ( C  e.  ( A  \  { B } )  <->  ( C  e.  A  /\  C  =/= 
B ) )
1713, 15, 16sylanbrc 413 . . 3  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  C  e.  ( A  \  { B } ) )
18 hashdifsn 10558 . . 3  |-  ( ( ( A  \  { B } )  e.  Fin  /\  C  e.  ( A 
\  { B }
) )  ->  ( `  ( ( A  \  { B } )  \  { C } ) )  =  ( ( `  ( A  \  { B }
) )  -  1 ) )
1912, 17, 18syl2anc 408 . 2  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  ( `  ( ( A  \  { B } )  \  { C } ) )  =  ( ( `  ( A  \  { B }
) )  -  1 ) )
20 hashdifsn 10558 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  ( `  ( A  \  { B } ) )  =  ( ( `  A )  -  1 ) )
21203ad2antr1 1146 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  ( `  ( A  \  { B } ) )  =  ( ( `  A
)  -  1 ) )
2221oveq1d 5782 . . 3  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  (
( `  ( A  \  { B } ) )  -  1 )  =  ( ( ( `  A
)  -  1 )  -  1 ) )
23 hashcl 10520 . . . . . 6  |-  ( A  e.  Fin  ->  ( `  A )  e.  NN0 )
2423nn0cnd 9025 . . . . 5  |-  ( A  e.  Fin  ->  ( `  A )  e.  CC )
25 sub1m1 8963 . . . . 5  |-  ( ( `  A )  e.  CC  ->  ( ( ( `  A
)  -  1 )  -  1 )  =  ( ( `  A
)  -  2 ) )
2624, 25syl 14 . . . 4  |-  ( A  e.  Fin  ->  (
( ( `  A
)  -  1 )  -  1 )  =  ( ( `  A
)  -  2 ) )
2726adantr 274 . . 3  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  (
( ( `  A
)  -  1 )  -  1 )  =  ( ( `  A
)  -  2 ) )
2822, 27eqtrd 2170 . 2  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  (
( `  ( A  \  { B } ) )  -  1 )  =  ( ( `  A
)  -  2 ) )
293, 19, 283eqtrd 2174 1  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  ( `  ( A  \  { B ,  C }
) )  =  ( ( `  A )  -  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 962    = wceq 1331    e. wcel 1480    =/= wne 2306    \ cdif 3063    C_ wss 3066   {csn 3522   {cpr 3523   ` cfv 5118  (class class class)co 5767   Fincfn 6627   CCcc 7611   1c1 7614    - cmin 7926   2c2 8764  ♯chash 10514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-frec 6281  df-1o 6306  df-oadd 6310  df-er 6422  df-en 6628  df-dom 6629  df-fin 6630  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-2 8772  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784  df-ihash 10515
This theorem is referenced by: (None)
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