Theorem ballotlemelo 34620

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 6838	. . 3 ⊢ (𝑑 = 𝐶 → ((♯‘𝑑) = 𝑀 ↔ (♯‘𝐶) = 𝑀))
2		ballotth.o	. . . 4 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
3		fveqeq2 6838	. . . . 5 ⊢ (𝑐 = 𝑑 → ((♯‘𝑐) = 𝑀 ↔ (♯‘𝑑) = 𝑀))
4	3	cbvrabv 3397	. . . 4 ⊢ {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀} = {𝑑 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑑) = 𝑀}
5	2, 4	eqtri 2758	. . 3 ⊢ 𝑂 = {𝑑 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑑) = 𝑀}
6	1, 5	elrab2 3634	. 2 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
7		ovex 7389	. . . 4 ⊢ (1...(𝑀 + 𝑁)) ∈ V
8	7	elpw2 5264	. . 3 ⊢ (𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝐶 ⊆ (1...(𝑀 + 𝑁)))
9	8	anbi1i 625	. 2 ⊢ ((𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀) ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))
10	6, 9	bitri 275	1 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (♯‘𝐶) = 𝑀))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3387   ⊆ wss 3885  𝒫 cpw 4531  ‘cfv 6487  (class class class)co 7356  1c1 11028   + caddc 11030  ℕcn 12163  ...cfz 13450  ♯chash 14281

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5220  ax-nul 5230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2931  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6443  df-fv 6495  df-ov 7359

This theorem is referenced by:  ballotlemscr  34651  ballotlemro  34655  ballotlemfg  34658  ballotlemfrc  34659  ballotlemfrceq  34661  ballotlemrinv0  34665