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Theorem ballotfilemofi 13200

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 9652	. . . . . 6 ⊢ 1 ∈ ℤ
3		ballotth.m	. . . . . . . 8 ⊢ 𝑀 ∈ ℕ
4		ballotth.n	. . . . . . . 8 ⊢ 𝑁 ∈ ℕ
5		nnaddcl 9306	. . . . . . . 8 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℕ)
6	3, 4, 5	mp2an 430	. . . . . . 7 ⊢ (𝑀 + 𝑁) ∈ ℕ
7	6	nnzi 9647	. . . . . 6 ⊢ (𝑀 + 𝑁) ∈ ℤ
8		fzfig 10848	. . . . . 6 ⊢ ((1 ∈ ℤ ∧ (𝑀 + 𝑁) ∈ ℤ) → (1...(𝑀 + 𝑁)) ∈ Fin)
9	2, 7, 8	mp2an 430	. . . . 5 ⊢ (1...(𝑀 + 𝑁)) ∈ Fin
10		fipwfi 7314	. . . . 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 11201	. . . . . . . . 9 ⊢ (𝑐 ∈ Fin → (♯‘𝑐) ∈ ℕ₀)
14	12, 13	syl 14	. . . . . . . 8 ⊢ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) → (♯‘𝑐) ∈ ℕ₀)
15	14	nn0zd 9748	. . . . . . 7 ⊢ (𝑐 ∈ (𝒫 (1...(𝑀 + 𝑁)) ∩ Fin) → (♯‘𝑐) ∈ ℤ)
16	3	nnzi 9647	. . . . . . 7 ⊢ 𝑀 ∈ ℤ
17		zdceq 9702	. . . . . . 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 7237	. . 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

Syntax hints: DECID wdc 846 = wceq 1402 ⊤wtru 1403 ∈ wcel 2209 ∀wral 2528 {crab 2532 ∩ cin 3219 𝒫 cpw 3688 ‘cfv 5375 (class class class)co 6078 Fincfn 7015 1c1 8173 + caddc 8175 ℕcn 9286 ℕ₀cn0 9545 ℤcz 9626 ...cfz 10393 ♯chash 11195

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 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 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8263 ax-resscn 8264 ax-1cn 8265 ax-1re 8266 ax-icn 8267 ax-addcl 8268 ax-addrcl 8269 ax-mulcl 8270 ax-addcom 8272 ax-addass 8274 ax-distr 8276 ax-i2m1 8277 ax-0lt1 8278 ax-0id 8280 ax-rnegex 8281 ax-cnre 8283 ax-pre-ltirr 8284 ax-pre-ltwlin 8285 ax-pre-lttrn 8286 ax-pre-apti 8287 ax-pre-ltadd 8288

This theorem 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 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-riota 6031 df-ov 6081 df-oprab 6082 df-mpo 6083 df-1st 6367 df-2nd 6368 df-recs 6569 df-frec 6655 df-1o 6680 df-2o 6681 df-er 6800 df-map 6917 df-en 7016 df-dom 7017 df-fin 7018 df-pnf 8355 df-mnf 8356 df-xr 8357 df-ltxr 8358 df-le 8359 df-sub 8492 df-neg 8493 df-inn 9287 df-n0 9546 df-z 9627 df-uz 9904 df-fz 10394 df-ihash 11196

This theorem is referenced by: ballotfilem2 13209 ballotfilemefi 13218 ballotfilemafi 13219 ballotfilembfi 13220 ballotfilem7 13260 ballotfilem8 13261 ballotfilemth 13262

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