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Theorem elopaelxp 5605
Description: Membership in an ordered-pair class abstraction implies membership in a Cartesian product. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
Assertion
Ref Expression
elopaelxp (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem elopaelxp
StepHypRef Expression
1 simpl 486 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝐴 = ⟨𝑥, 𝑦⟩)
212eximi 1837 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
3 elopab 5379 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
4 elvv 5590 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
52, 3, 43imtr4i 295 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2111  Vcvv 3441  cop 4531  {copab 5092   × cxp 5517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-xp 5525
This theorem is referenced by:  bropaex12  5606  clwlkcompim  27569  linedegen  33717  opelopab3  35155
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