Theorem ballotlemelo 31091

Description: Elementhood in 𝑂. (Contributed by Thierry Arnoux, 17-Apr-2017.)

Hypotheses

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

Assertion

Ref	Expression
ballotlemelo	⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))

Distinct variable groups:   𝑀,𝑐   𝑁,𝑐   𝑂,𝑐

Allowed substitution hint:   𝐶(𝑐)

Proof of Theorem ballotlemelo

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveqeq2 6446	. . 3 ⊢ (𝑑 = 𝐶 → ((♯‘𝑑) = 𝑀 ↔ (♯‘𝐶) = 𝑀))
2		ballotth.o	. . . 4 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
3		fveqeq2 6446	. . . . 5 ⊢ (𝑐 = 𝑑 → ((♯‘𝑐) = 𝑀 ↔ (♯‘𝑑) = 𝑀))
4	3	cbvrabv 3412	. . . 4 ⊢ {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} = {𝑑 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑑) = 𝑀}
5	2, 4	eqtri 2849	. . 3 ⊢ 𝑂 = {𝑑 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑑) = 𝑀}
6	1, 5	elrab2 3589	. 2 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
7		ovex 6942	. . . 4 ⊢ (1...(𝑀 + 𝑁)) ∈ V
8	7	elpw2 5052	. . 3 ⊢ (𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝐶 ⊆ (1...(𝑀 + 𝑁)))
9	8	anbi1i 617	. 2 ⊢ ((𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀) ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
10	6, 9	bitri 267	1 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))

Syntax hints:   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164  {crab 3121   ⊆ wss 3798  𝒫 cpw 4380  ‘cfv 6127  (class class class)co 6910  1c1 10260   + caddc 10262  ℕcn 11357  ...cfz 12626  ♯chash 13417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913

This theorem is referenced by:  ballotlemscr  31122  ballotlemro  31126  ballotlemfg  31129  ballotlemfrc  31130  ballotlemfrceq  31132  ballotlemrinv0  31136