Theorem ballotlemoex 34484

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 7423	. . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ V
3	2	pwex 5338	. 2 ⊢ 𝒫 (1...(𝑀 + 𝑁)) ∈ V
4	1, 3	rabex2 5299	1 ⊢ 𝑂 ∈ V

Syntax hints:   = wceq 1540   ∈ wcel 2109  {crab 3408  Vcvv 3450  𝒫 cpw 4566  ‘cfv 6514  (class class class)co 7390  1c1 11076   + caddc 11078  ℕcn 12193  ...cfz 13475  ♯chash 14302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-pw 4568  df-sn 4593  df-pr 4595  df-uni 4875  df-iota 6467  df-fv 6522  df-ov 7393

This theorem is referenced by:  ballotlem2  34487  ballotlem8  34535