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Theorem elopaba 5155
 Description: Membership in an ordered pair class builder. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
copsex2ga.1 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
Assertion
Ref Expression
elopaba (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem elopaba
StepHypRef Expression
1 elopab 4908 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
2 copsex2ga.1 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
32copsex2gb 5153 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
41, 3bitri 263 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  {copab 4642   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044 This theorem is referenced by:  dicelvalN  35485
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