Theorem elopabr 5544

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 5174	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} ⊆ 𝑅
2	1	sseli 3932	1 ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} → 𝐴 ∈ 𝑅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142   class class class wbr 5108  {copab 5172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ss 3921  df-br 5109  df-opab 5173

This theorem is used by: (None)