ballotfilemofi - Intuitionistic Logic Explorer

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Theorem ballotfilemofi 13219

Description: 𝑂 is finite. (Contributed by Jim Kingdon, 20-May-2026.)

**Hypotheses**
Ref	Expression
ballotth.m	⊢ 𝑀 ∈ ℕ
ballotth.n	⊢ 𝑁 ∈ ℕ
ballotfilem.o	⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}

**Assertion**
Ref	Expression
ballotfilemofi	⊢ 𝑂 ∈ Fin

Distinct variable groups: 𝑀,𝑐 𝑁,𝑐 𝑂,𝑐

**Proof of Theorem ballotfilemofi**
Step	Hyp	Ref	Expression
1		ballotfilem.o	. 2 ⊢ 𝑂 = {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀}
2		1z 9670	. . . . . 6 ⊢ 1 ∈ ℤ
3		ballotth.m	. . . . . . . 8 ⊢ 𝑀 ∈ ℕ
4		ballotth.n	. . . . . . . 8 ⊢ 𝑁 ∈ ℕ
5		nnaddcl 9324	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
6	3, 4, 5	mp2an 430	. . . . . . 7 ⊢ (𝑀 + 𝑁) ∈ ℕ
7	6	nnzi 9665	. . . . . 6 ⊢ (𝑀 + 𝑁) ∈ ℤ
8		fzfig 10867	. . . . . 6 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (1...(𝑀 + 𝑁)) ∈ Fin)
9	2, 7, 8	mp2an 430	. . . . 5 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
10		fipwfi 7321	. . . . 5 ⊢ ((1...(𝑀 + 𝑁)) ∈ Fin → (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∈ Fin)
11	9, 10	mp1i 10	. . . 4 ⊢ (⊤ → (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∈ Fin)
12		elinel2 3416	. . . . . . . . 9 ⊢ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) → 𝑐 ∈ Fin)
13		hashcl 11220	. . . . . . . . 9 ⊢ (𝑐 ∈ Fin → (♯‘𝑐) ∈ ℕ₀)
14	12, 13	syl 14	. . . . . . . 8 ⊢ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) → (♯‘𝑐) ∈ ℕ₀)
15	14	nn0zd 9766	. . . . . . 7 ⊢ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) → (♯‘𝑐) ∈ ℤ)
16	3	nnzi 9665	. . . . . . 7 ⊢ 𝑀 ∈ ℤ
17		zdceq 9720	. . . . . . 7 ⊢ (((♯‘𝑐) ∈ ℤ ∧ 𝑀 ∈ ℤ) → DECID (♯‘𝑐) = 𝑀)
18	15, 16, 17	sylancl 417	. . . . . 6 ⊢ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) → DECID (♯‘𝑐) = 𝑀)
19	18	rgen 2603	. . . . 5 ⊢ ∀𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin)DECID (♯‘𝑐) = 𝑀
20	19	a1i 9	. . . 4 ⊢ (⊤ → ∀𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin)DECID (♯‘𝑐) = 𝑀)
21	11, 20	ssfirab 7244	. . 3 ⊢ (⊤ → {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀} ∈ Fin)
22	21	mptru 1411	. 2 ⊢ {𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) ∣ (♯‘𝑐) = 𝑀} ∈ Fin
23	1, 22	eqeltri 2311	1 ⊢ 𝑂 ∈ Fin

Colors of variables: wff set class

This proof depends on syntax axioms: DECID wdc 846 = wceq 1402 ⊤wtru 1403 ∈ wcel 2209 ∀wral 2528 {crab 2532 ∩ cin 3219 𝒫 cpw 3688 ‘cfv 5377 (class class class)co 6085 Fincfn 7022 1c1 8180 + caddc 8182 ℕcn 9304 ℕ₀cn0 9563 ℤcz 9644 ...cfz 10411 ♯chash 11214

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-13 2211 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-addass 8281 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-1o 6687 df-2o 6688 df-er 6807 df-map 6924 df-en 7023 df-dom 7024 df-fin 7025 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-inn 9305 df-n0 9564 df-z 9645 df-uz 9922 df-fz 10412 df-ihash 11215

This theorem is used by: ballotfilem2 13228 ballotfilemefi 13237 ballotfilemafi 13238 ballotfilembfi 13239 ballotfilem7 13279 ballotfilem8 13280 ballotfilemth 13281

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