Theorem elopabr 5563

Description: Membership in an ordered-pair class abstraction defined by a binary relation. (Contributed by AV, 16-Feb-2021.) (Proof shortened by SN, 11-Dec-2024.)

Assertion

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

Distinct variable groups:   𝑥,𝑅   𝑦,𝑅

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem elopabr

Step	Hyp	Ref	Expression
1		opabss 5212	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
2	1	sseli 3976	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2099   class class class wbr 5148  {copab 5210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-in 3954  df-ss 3964  df-br 5149  df-opab 5211

This theorem is referenced by: (None)