Metamath Proof Explorer

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 opabss 
 |-  { <. x , y >. | x R y } C_ R
4 2 3 sstri 
 |-  { <. x , y >. | ( x R y /\ ps ) } C_ R
5 4 sseli 
 |-  ( A e. { <. x , y >. | ( x R y /\ ps ) } -> A e. R )