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Theorem hashdifpr 13772

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 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = ((♯‘𝐴) − 2))

**Proof of Theorem hashdifpr**
Step	Hyp	Ref	Expression
1		difpr 4696	. . . 4 ⊢ (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶})
2	1	a1i 11	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (𝐴 ∖ {𝐵, 𝐶}) = ((𝐴 ∖ {𝐵}) ∖ {𝐶}))
3	2	fveq2d 6649	. 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = (♯‘((𝐴 ∖ {𝐵}) ∖ {𝐶})))
4		diffi 8734	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝐵}) ∈ Fin)
5		necom 3040	. . . . . . . 8 ⊢ (𝐵 ≠ 𝐶 ↔ 𝐶 ≠ 𝐵)
6	5	biimpi 219	. . . . . . 7 ⊢ (𝐵 ≠ 𝐶 → 𝐶 ≠ 𝐵)
7	6	anim2i 619	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶) → (𝐶 ∈ 𝐴 ∧ 𝐶 ≠ 𝐵))
8	7	3adant1 1127	. . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶) → (𝐶 ∈ 𝐴 ∧ 𝐶 ≠ 𝐵))
9	8	adantl 485	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (𝐶 ∈ 𝐴 ∧ 𝐶 ≠ 𝐵))
10		eldifsn 4680	. . . 4 ⊢ (𝐶 ∈ (𝐴 ∖ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ 𝐶 ≠ 𝐵))
11	9, 10	sylibr 237	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → 𝐶 ∈ (𝐴 ∖ {𝐵}))
12		hashdifsn 13771	. . 3 ⊢ (((𝐴 ∖ {𝐵}) ∈ Fin ∧ 𝐶 ∈ (𝐴 ∖ {𝐵})) → (♯‘((𝐴 ∖ {𝐵}) ∖ {𝐶})) = ((♯‘(𝐴 ∖ {𝐵})) − 1))
13	4, 11, 12	syl2an2r 684	. 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘((𝐴 ∖ {𝐵}) ∖ {𝐶})) = ((♯‘(𝐴 ∖ {𝐵})) − 1))
14		hashdifsn 13771	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → (♯‘(𝐴 ∖ {𝐵})) = ((♯‘𝐴) − 1))
15	14	3ad2antr1 1185	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵})) = ((♯‘𝐴) − 1))
16	15	oveq1d 7150	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → ((♯‘(𝐴 ∖ {𝐵})) − 1) = (((♯‘𝐴) − 1) − 1))
17		hashcl 13713	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
18	17	nn0cnd 11945	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
19		sub1m1 11877	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℂ → (((♯‘𝐴) − 1) − 1) = ((♯‘𝐴) − 2))
20	18, 19	syl 17	. . . 4 ⊢ (𝐴 ∈ Fin → (((♯‘𝐴) − 1) − 1) = ((♯‘𝐴) − 2))
21	20	adantr 484	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (((♯‘𝐴) − 1) − 1) = ((♯‘𝐴) − 2))
22	16, 21	eqtrd 2833	. 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → ((♯‘(𝐴 ∖ {𝐵})) − 1) = ((♯‘𝐴) − 2))
23	3, 13, 22	3eqtrd 2837	1 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ≠ 𝐶)) → (♯‘(𝐴 ∖ {𝐵, 𝐶})) = ((♯‘𝐴) − 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 {csn 4525 {cpr 4527 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 1c1 10527 − cmin 10859 2c2 11680 ♯chash 13686

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687

This theorem is referenced by: nbfusgrlevtxm2 27168

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