Theorem elopabr

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)

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

Proof

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