Theorem ballotlemoex 31747

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 7192	. . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ V
3	2	pwex 5284	. 2 ⊢ 𝒫 (1...(𝑀 + 𝑁)) ∈ V
4	1, 3	rabex2 5240	1 ⊢ 𝑂 ∈ V

Syntax hints:   = wceq 1536   ∈ wcel 2113  {crab 3145  Vcvv 3497  𝒫 cpw 4542  ‘cfv 6358  (class class class)co 7159  1c1 10541   + caddc 10543  ℕcn 11641  ...cfz 12895  ♯chash 13693

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-pw 4544  df-sn 4571  df-pr 4573  df-uni 4842  df-iota 6317  df-fv 6366  df-ov 7162

This theorem is referenced by:  ballotlem2  31750  ballotlem8  31798