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Theorem ballotlemelo 30330
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 (𝐶𝑂 ↔ (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.
StepHypRef Expression
1 fveq2 6148 . . . 4 (𝑑 = 𝐶 → (#‘𝑑) = (#‘𝐶))
21eqeq1d 2623 . . 3 (𝑑 = 𝐶 → ((#‘𝑑) = 𝑀 ↔ (#‘𝐶) = 𝑀))
3 ballotth.o . . . 4 𝑂 = {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀}
4 fveq2 6148 . . . . . 6 (𝑐 = 𝑑 → (#‘𝑐) = (#‘𝑑))
54eqeq1d 2623 . . . . 5 (𝑐 = 𝑑 → ((#‘𝑐) = 𝑀 ↔ (#‘𝑑) = 𝑀))
65cbvrabv 3185 . . . 4 {𝑐 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑐) = 𝑀} = {𝑑 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑑) = 𝑀}
73, 6eqtri 2643 . . 3 𝑂 = {𝑑 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∣ (#‘𝑑) = 𝑀}
82, 7elrab2 3348 . 2 (𝐶𝑂 ↔ (𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀))
9 ovex 6632 . . . 4 (1...(𝑀 + 𝑁)) ∈ V
109elpw2 4788 . . 3 (𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ↔ 𝐶 ⊆ (1...(𝑀 + 𝑁)))
1110anbi1i 730 . 2 ((𝐶 ∈ 𝒫 (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀) ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀))
128, 11bitri 264 1 (𝐶𝑂 ↔ (𝐶 ⊆ (1...(𝑀 + 𝑁)) ∧ (#‘𝐶) = 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  {crab 2911  wss 3555  𝒫 cpw 4130  cfv 5847  (class class class)co 6604  1c1 9881   + caddc 9883  cn 10964  ...cfz 12268  #chash 13057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607
This theorem is referenced by:  ballotlemscr  30361  ballotlemro  30365  ballotlemfg  30368  ballotlemfrc  30369  ballotlemfrceq  30371  ballotlemrinv0  30375
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