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Metamath Proof Explorer

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	$⊢ A \in \{⟨x, y⟩ \| x R y\} \to A \in R$

Proof

Step	Hyp	Ref	Expression
1		opabss	$⊢ \{⟨x, y⟩ \| x R y\} \subseteq R$
2	1	sseli	$⊢ A \in \{⟨x, y⟩ \| x R y\} \to A \in R$