Theorem ballotlemelo 30549

Description: Elementhood in 𝑂. (Contributed by Thierry Arnoux, 17-Apr-2017.)

Hypotheses

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

Assertion

Ref	Expression
ballotlemelo	⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀))

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

Allowed substitution hint:   𝐶(𝑐)

Proof of Theorem ballotlemelo

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 ⊢ (𝑑 = 𝐶 → (#‘𝑑) = (#‘𝐶))
2	1	eqeq1d 2624	. . 3 ⊢ (𝑑 = 𝐶 → ((#‘𝑑) = 𝑀 ↔ (#‘𝐶) = 𝑀))
3		ballotth.o	. . . 4 ⊢ 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}
4		fveq2 6191	. . . . . 6 ⊢ (𝑐 = 𝑑 → (#‘𝑐) = (#‘𝑑))
5	4	eqeq1d 2624	. . . . 5 ⊢ (𝑐 = 𝑑 → ((#‘𝑐) = 𝑀 ↔ (#‘𝑑) = 𝑀))
6	5	cbvrabv 3199	. . . 4 ⊢ {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀} = {𝑑 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑑) = 𝑀}
7	3, 6	eqtri 2644	. . 3 ⊢ 𝑂 = {𝑑 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑑) = 𝑀}
8	2, 7	elrab2 3366	. 2 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀))
9		ovex 6678	. . . 4 ⊢ (1...(𝑀 + 𝑁)) ∈ V
10	9	elpw2 4828	. . 3 ⊢ (𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝐶 ⊆ (1...(𝑀 + 𝑁)))
11	10	anbi1i 731	. 2 ⊢ ((𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀) ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀))
12	8, 11	bitri 264	1 ⊢ (𝐶 ∈ 𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916   ⊆ wss 3574  𝒫 cpw 4158  ‘cfv 5888  (class class class)co 6650  1c1 9937   + caddc 9939  ℕcn 11020  ...cfz 12326  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  ballotlemscr  30580  ballotlemro  30584  ballotlemfg  30587  ballotlemfrc  30588  ballotlemfrceq  30590  ballotlemrinv0  30594