Theorem elopaba

Description: Membership in an ordered-pair class abstraction. (Contributed by NM, 25-Feb-2014) (Revised by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Hypothesis	copsex2ga.1	|- ( A = <. x , y >. -> ( ph <-> ps ) )
Assertion	elopaba	|- ( A e. { <. x , y >. | ps } <-> ( A e. ( _V X. _V ) /\ ph ) )

Proof

Step	Hyp	Ref	Expression
1		copsex2ga.1	|- ( A = <. x , y >. -> ( ph <-> ps ) )
2		elopab	|- ( A e. { <. x , y >. | ps } <-> E. x E. y ( A = <. x , y >. /\ ps ) )
3	1	copsex2gb	|- ( E. x E. y ( A = <. x , y >. /\ ps ) <-> ( A e. ( _V X. _V ) /\ ph ) )
4	2 3	bitri	|- ( A e. { <. x , y >. | ps } <-> ( A e. ( _V X. _V ) /\ ph ) )