Theorem elopabrOLD 5556

Description: Obsolete version of elopabr 5554 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 5520	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦))
2		df-br 5142	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
3	2	biimpi 215	. . . . 5 ⊢ (𝑥𝑅𝑦 → ⟨𝑥, 𝑦⟩ ∈ 𝑅)
4		eleq1 2815	. . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝐴 ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
5	3, 4	imbitrrid 245	. . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (𝑥𝑅𝑦 → 𝐴 ∈ 𝑅))
6	5	imp 406	. . 3 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅)
7	6	exlimivv 1927	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑥𝑅𝑦) → 𝐴 ∈ 𝑅)
8	1, 7	sylbi 216	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ⟨cop 4629   class class class wbr 5141  {copab 5203

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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204

This theorem is referenced by: (None)