Theorem elopaelxpOLD 5764

Description: Obsolete version of elopaelxp 5763 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

Step	Hyp	Ref	Expression
1		simpl 483	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → 𝐴 = ⟨𝑥, 𝑦⟩)
2	1	2eximi 1838	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) → ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
3		elopab 5526	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
4		elvv 5748	. 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
5	2, 3, 4	3imtr4i 291	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜓} → 𝐴 ∈ (V × V))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3474  ⟨cop 4633  {copab 5209   × cxp 5673

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-opab 5210  df-xp 5681

This theorem is referenced by: (None)