Theorem elopabran

Description: Membership in an ordered-pair class abstraction defined by a restricted binary relation. (Contributed by AV, 16-Feb-2021)

	Ref	Expression
Assertion	elopabran	|- ( A e. { <. x , y >. | ( x R y /\ ps ) } -> A e. R )

Proof

Step	Hyp	Ref	Expression
1		simpl	|- ( ( x R y /\ ps ) -> x R y )
2	1	ssopab2i	|- { <. x , y >. | ( x R y /\ ps ) } C_ { <. x , y >. | x R y }
3	2	sseli	|- ( A e. { <. x , y >. | ( x R y /\ ps ) } -> A e. { <. x , y >. | x R y } )
4		elopabr	|- ( A e. { <. x , y >. | x R y } -> A e. R )
5	3 4	syl	|- ( A e. { <. x , y >. | ( x R y /\ ps ) } -> A e. R )