Theorem ballotlemoex 34943

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 7452	. . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ V
3	2	pwex 5353	. 2 ⊢ 𝒫 (1...(𝑀 + 𝑁)) ∈ V
4	1, 3	rabex2 5313	1 ⊢ 𝑂 ∈ V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  {crab 3418  Vcvv 3457  𝒫 cpw 4564  ‘cfv 6540  (class class class)co 7419  1c1 11118   + caddc 11120  ℕcn 12250  ...cfz 13553  ♯chash 14386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-pw 4566  df-sn 4592  df-pr 4594  df-uni 4875  df-iota 6496  df-fv 6548  df-ov 7422

This theorem is used by:  ballotlem2  34946  ballotlem8  34994