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Mirrors > Home > ILE Home > Th. List > hashdifpr | GIF version |
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.) |
Ref | Expression |
---|---|
hashdifpr | ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = ((♯‘𝐴) − 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difpr 3736 | . . . 4 ⊢ (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶}) | |
2 | 1 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})) |
3 | 2 | fveq2d 5521 | . 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = (♯‘((𝐴 ∖ {𝐵}) ∖ {𝐶}))) |
4 | simpl 109 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → 𝐴 ∈ Fin) | |
5 | snfig 6816 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ∈ Fin) | |
6 | 5 | 3ad2ant1 1018 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶) → {𝐵} ∈ Fin) |
7 | 6 | adantl 277 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → {𝐵} ∈ Fin) |
8 | snssi 3738 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴) | |
9 | 8 | 3ad2ant1 1018 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶) → {𝐵} ⊆ 𝐴) |
10 | 9 | adantl 277 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → {𝐵} ⊆ 𝐴) |
11 | diffifi 6896 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝐵} ∈ Fin ∧ {𝐵} ⊆ 𝐴) → (𝐴 ∖ {𝐵}) ∈ Fin) | |
12 | 4, 7, 10, 11 | syl3anc 1238 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (𝐴 ∖ {𝐵}) ∈ Fin) |
13 | simpr2 1004 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → 𝐶 ∈ 𝐴) | |
14 | simpr3 1005 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → 𝐵 ≠ 𝐶) | |
15 | 14 | necomd 2433 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → 𝐶 ≠ 𝐵) |
16 | eldifsn 3721 | . . . 4 ⊢ (𝐶 ∈ (𝐴 ∖ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ 𝐶 ≠ 𝐵)) | |
17 | 13, 15, 16 | sylanbrc 417 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → 𝐶 ∈ (𝐴 ∖ {𝐵})) |
18 | hashdifsn 10801 | . . 3 ⊢ (((𝐴 ∖ {𝐵}) ∈ Fin ∧ 𝐶 ∈ (𝐴 ∖ {𝐵})) → (♯‘((𝐴 ∖ {𝐵}) ∖ {𝐶})) = ((♯‘(𝐴 ∖ {𝐵})) − 1)) | |
19 | 12, 17, 18 | syl2anc 411 | . 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘((𝐴 ∖ {𝐵}) ∖ {𝐶})) = ((♯‘(𝐴 ∖ {𝐵})) − 1)) |
20 | hashdifsn 10801 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → (♯‘(𝐴 ∖ {𝐵})) = ((♯‘𝐴) − 1)) | |
21 | 20 | 3ad2antr1 1162 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵})) = ((♯‘𝐴) − 1)) |
22 | 21 | oveq1d 5892 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → ((♯‘(𝐴 ∖ {𝐵})) − 1) = (((♯‘𝐴) − 1) − 1)) |
23 | hashcl 10763 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
24 | 23 | nn0cnd 9233 | . . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ) |
25 | sub1m1 9171 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℂ → (((♯‘𝐴) − 1) − 1) = ((♯‘𝐴) − 2)) | |
26 | 24, 25 | syl 14 | . . . 4 ⊢ (𝐴 ∈ Fin → (((♯‘𝐴) − 1) − 1) = ((♯‘𝐴) − 2)) |
27 | 26 | adantr 276 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (((♯‘𝐴) − 1) − 1) = ((♯‘𝐴) − 2)) |
28 | 22, 27 | eqtrd 2210 | . 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → ((♯‘(𝐴 ∖ {𝐵})) − 1) = ((♯‘𝐴) − 2)) |
29 | 3, 19, 28 | 3eqtrd 2214 | 1 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = ((♯‘𝐴) − 2)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∖ cdif 3128 ⊆ wss 3131 {csn 3594 {cpr 3595 ‘cfv 5218 (class class class)co 5877 Fincfn 6742 ℂcc 7811 1c1 7814 − cmin 8130 2c2 8972 ♯chash 10757 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-frec 6394 df-1o 6419 df-oadd 6423 df-er 6537 df-en 6743 df-dom 6744 df-fin 6745 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-inn 8922 df-2 8980 df-n0 9179 df-z 9256 df-uz 9531 df-fz 10011 df-ihash 10758 |
This theorem is referenced by: (None) |
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