Theorem elopabr 5548

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 5189	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
2	1	sseli 3961	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅)

Syntax hints:   → wi 4   ∈ wcel 2107   class class class wbr 5125  {copab 5187

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 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ss 3950  df-br 5126  df-opab 5188

This theorem is referenced by: (None)