Theorem elopabrOLD 5567

Description: Obsolete version of elopabr 5565 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 5531	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
2		df-br 5143	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
3	2	biimpi 216	. . . . 5 ⊢ (𝑥𝑅𝑦 → ⟨𝑥, 𝑦⟩ ∈ 𝑅)
4		eleq1 2828	. . . . 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 1931	. 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 1539  ∃wex 1778   ∈ wcel 2107  ⟨cop 4631   class class class wbr 5142  {copab 5204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205

This theorem is referenced by: (None)