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Theorem elopaelxpOLD 5763
Description: Obsolete version of elopaelxp 5762 as of 11-Dec-2024. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
elopaelxpOLD (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem elopaelxpOLD
StepHypRef Expression
1 simpl 481 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝐴 = ⟨𝑥, 𝑦⟩)
212eximi 1830 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
3 elopab 5524 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
4 elvv 5747 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
52, 3, 43imtr4i 291 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3463  cop 4631  {copab 5206   × cxp 5671
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 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
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 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-dif 3944  df-un 3946  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5207  df-xp 5679
This theorem is referenced by: (None)
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