Theorem ballotlemoex 34450

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 7481	. . 3 ⊢ (1...(𝑀 + 𝑁)) ∈ V
3	2	pwex 5398	. 2 ⊢ 𝒫 (1...(𝑀 + 𝑁)) ∈ V
4	1, 3	rabex2 5359	1 ⊢ 𝑂 ∈ V

Syntax hints:   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488  𝒫 cpw 4622  ‘cfv 6573  (class class class)co 7448  1c1 11185   + caddc 11187  ℕcn 12293  ...cfz 13567  ♯chash 14379

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-pw 4624  df-sn 4649  df-pr 4651  df-uni 4932  df-iota 6525  df-fv 6581  df-ov 7451

This theorem is referenced by:  ballotlem2  34453  ballotlem8  34501