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Theorem hashdifpr 10819
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 3749 . . . 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 5534 . 2  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  ( `  ( A  \  { B ,  C }
) )  =  ( `  ( ( A  \  { B } )  \  { C } ) ) )
4 simpl 109 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  A  e.  Fin )
5 snfig 6832 . . . . . 6  |-  ( B  e.  A  ->  { B }  e.  Fin )
653ad2ant1 1020 . . . . 5  |-  ( ( B  e.  A  /\  C  e.  A  /\  B  =/=  C )  ->  { B }  e.  Fin )
76adantl 277 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  { B }  e.  Fin )
8 snssi 3751 . . . . . 6  |-  ( B  e.  A  ->  { B }  C_  A )
983ad2ant1 1020 . . . . 5  |-  ( ( B  e.  A  /\  C  e.  A  /\  B  =/=  C )  ->  { B }  C_  A
)
109adantl 277 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  { B }  C_  A )
11 diffifi 6912 . . . 4  |-  ( ( A  e.  Fin  /\  { B }  e.  Fin  /\ 
{ B }  C_  A )  ->  ( A  \  { B }
)  e.  Fin )
124, 7, 10, 11syl3anc 1249 . . 3  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  ( A  \  { B }
)  e.  Fin )
13 simpr2 1006 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  C  e.  A )
14 simpr3 1007 . . . . 5  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  B  =/=  C )
1514necomd 2446 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  C  =/=  B )
16 eldifsn 3734 . . . 4  |-  ( C  e.  ( A  \  { B } )  <->  ( C  e.  A  /\  C  =/= 
B ) )
1713, 15, 16sylanbrc 417 . . 3  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  C  e.  ( A  \  { B } ) )
18 hashdifsn 10818 . . 3  |-  ( ( ( A  \  { B } )  e.  Fin  /\  C  e.  ( A 
\  { B }
) )  ->  ( `  ( ( A  \  { B } )  \  { C } ) )  =  ( ( `  ( A  \  { B }
) )  -  1 ) )
1912, 17, 18syl2anc 411 . 2  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  ( `  ( ( A  \  { B } )  \  { C } ) )  =  ( ( `  ( A  \  { B }
) )  -  1 ) )
20 hashdifsn 10818 . . . . 5  |-  ( ( A  e.  Fin  /\  B  e.  A )  ->  ( `  ( A  \  { B } ) )  =  ( ( `  A )  -  1 ) )
21203ad2antr1 1164 . . . 4  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  ( `  ( A  \  { B } ) )  =  ( ( `  A
)  -  1 ) )
2221oveq1d 5906 . . 3  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  (
( `  ( A  \  { B } ) )  -  1 )  =  ( ( ( `  A
)  -  1 )  -  1 ) )
23 hashcl 10780 . . . . . 6  |-  ( A  e.  Fin  ->  ( `  A )  e.  NN0 )
2423nn0cnd 9250 . . . . 5  |-  ( A  e.  Fin  ->  ( `  A )  e.  CC )
25 sub1m1 9188 . . . . 5  |-  ( ( `  A )  e.  CC  ->  ( ( ( `  A
)  -  1 )  -  1 )  =  ( ( `  A
)  -  2 ) )
2624, 25syl 14 . . . 4  |-  ( A  e.  Fin  ->  (
( ( `  A
)  -  1 )  -  1 )  =  ( ( `  A
)  -  2 ) )
2726adantr 276 . . 3  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  (
( ( `  A
)  -  1 )  -  1 )  =  ( ( `  A
)  -  2 ) )
2822, 27eqtrd 2222 . 2  |-  ( ( A  e.  Fin  /\  ( B  e.  A  /\  C  e.  A  /\  B  =/=  C
) )  ->  (
( `  ( A  \  { B } ) )  -  1 )  =  ( ( `  A
)  -  2 ) )
293, 19, 283eqtrd 2226 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 104    /\ w3a 980    = wceq 1364    e. wcel 2160    =/= wne 2360    \ cdif 3141    C_ wss 3144   {csn 3607   {cpr 3608   ` cfv 5231  (class class class)co 5891   Fincfn 6758   CCcc 7828   1c1 7831    - cmin 8147   2c2 8989  ♯chash 10774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-addcom 7930  ax-addass 7932  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-0id 7938  ax-rnegex 7939  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-frec 6410  df-1o 6435  df-oadd 6439  df-er 6553  df-en 6759  df-dom 6760  df-fin 6761  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-inn 8939  df-2 8997  df-n0 9196  df-z 9273  df-uz 9548  df-fz 10028  df-ihash 10775
This theorem is referenced by: (None)
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