Theorem elopabrOLD 5523

Description: Obsolete version of elopabr 5521 as of 11-Dec-2024. (Contributed by AV, 16-Feb-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elopabrOLD	⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦

Proof of Theorem elopabrOLD

Step	Hyp	Ref	Expression
1		elopab 5487	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
2		df-br 5108	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
3	2	biimpi 216	. . . . 5 ⊢ (𝑥𝑅𝑦 → ⟨𝑥, 𝑦⟩ ∈ 𝑅)
4		eleq1 2816	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
5	3, 4	imbitrrid 246	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥𝑅𝑦 → 𝐴 ∈ 𝑅))
6	5	imp 406	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅)
7	6	exlimivv 1932	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅)
8	1, 7	sylbi 217	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ⟨cop 4595   class class class wbr 5107  {copab 5169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170

This theorem is referenced by: (None)