Theorem ballotlemoex 35171

Description: 𝑂 is a set. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Hypotheses

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

Assertion

Ref	Expression
ballotlemoex	⊢ 𝑂 ∈ V

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

Proof of Theorem ballotlemoex

Step	Hyp	Ref	Expression
1		ballotth.o	. 2 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (♯‘𝑐) = 𝑀}
2		ovex 7460	. . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ V
3	2	pwex 5345	. 2 ⊢ 𝒫 (1...(𝑀 + 𝑁)) ∈ V
4	1, 3	rabex2 5305	1 ⊢ 𝑂 ∈ V

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2146  {crab 3415  Vcvv 3453  𝒫 cpw 4559  ‘cfv 6546  (class class class)co 7427  1c1 11211   + caddc 11213  ℕcn 12345  ...cfz 13649  ♯chash 14484

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-pw 4561  df-sn 4588  df-pr 4590  df-uni 4870  df-iota 6500  df-fv 6554  df-ov 7430

This theorem is used by:  ballotlem2  35174  ballotlem8  35222