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Theorem hashdifpr 11183
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 3835 . . . 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 5673 . 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 7055 . . . . . 6 |- ( B e. A -> { B } e. Fin )
653ad2ant1 1045 . . . . 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 3837 . . . . . 6 |- ( B e. A -> { B } C_ A )
983ad2ant1 1045 . . . . 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 7150 . . . 4 |- ( ( A e. Fin /\ { B } e. Fin /\ { B } C_ A ) -> ( A \ { B } ) e. Fin )
124, 7, 10, 11syl3anc 1274 . . 3 |- ( ( A e. Fin /\ ( B e. A /\ C e. A /\ B =/= C ) ) -> ( A \ { B } ) e. Fin )
13 simpr2 1031 . . . 4 |- ( ( A e. Fin /\ ( B e. A /\ C e. A /\ B =/= C ) ) -> C e. A )
14 simpr3 1032 . . . . 5 |- ( ( A e. Fin /\ ( B e. A /\ C e. A /\ B =/= C ) ) -> B =/= C )
1514necomd 2498 . . . 4 |- ( ( A e. Fin /\ ( B e. A /\ C e. A /\ B =/= C ) ) -> C =/= B )
16 eldifsn 3819 . . . 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 11182 . . 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 11182 . . . . 5 |- ( ( A e. Fin /\ B e. A ) -> (♯ ` ( A \ { B } ) ) = ( (♯ ` A ) - 1 ) )
21203ad2antr1 1189 . . . 4 |- ( ( A e. Fin /\ ( B e. A /\ C e. A /\ B =/= C ) ) -> (♯ ` ( A \ { B } ) ) = ( (♯ ` A ) - 1 ) )
2221oveq1d 6064 . . 3 |- ( ( A e. Fin /\ ( B e. A /\ C e. A /\ B =/= C ) ) -> ( (♯ ` ( A \ { B } ) ) - 1 ) = ( ( (♯ ` A ) - 1 ) - 1 ) )
23 hashcl 11142 . . . . . 6 |- ( A e. Fin -> (♯ ` A ) e. NN0 )
2423nn0cnd 9554 . . . . 5 |- ( A e. Fin -> (♯ ` A ) e. CC )
25 sub1m1 9488 . . . . 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 2265 . 2 |- ( ( A e. Fin /\ ( B e. A /\ C e. A /\ B =/= C ) ) -> ( (♯ ` ( A \ { B } ) ) - 1 ) = ( (♯ ` A ) - 2 ) )
293, 19, 283eqtrd 2269 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 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   \ cdif 3207   C_ wss 3210   {csn 3688   {cpr 3689   ` cfv 5351  (class class class)co 6049   Fincfn 6974   CCcc 8124   1c1 8127   - cmin 8443   2c2 9287  ♯chash 11136
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-oadd 6650  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-2 9295  df-n0 9496  df-z 9577  df-uz 9853  df-fz 10342  df-ihash 11137
This theorem is referenced by: (None)
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