Theorem elscottab

Description: An element of the output of the Scott operation applied to a class abstraction satisfies the class abstraction's predicate. (Contributed by Rohan Ridenour, 14-Aug-2023)

	Ref	Expression
Hypothesis	elscottab.1	|- ( x = y -> ( ph <-> ps ) )
Assertion	elscottab	|- ( y e. Scott { x | ph } -> ps )

Proof

Step	Hyp	Ref	Expression
1		elscottab.1	|- ( x = y -> ( ph <-> ps ) )
2		scottss	|- Scott { x | ph } C_ { x | ph }
3	2	sseli	|- ( y e. Scott { x | ph } -> y e. { x | ph } )
4		vex	|- y e. _V
5	4 1	elab	|- ( y e. { x | ph } <-> ps )
6	3 5	sylib	|- ( y e. Scott { x | ph } -> ps )