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Theorem elopaba 5812
Description: Membership in an ordered-pair class abstraction. (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 5531 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
2 copsex2ga.1 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
32copsex2gb 5810 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
41, 3bitri 274 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3471  cop 4636  {copab 5212   × cxp 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5213  df-xp 5686
This theorem is referenced by:  dicelvalN  40655
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