Theorem neicvgrcomplex

Description: The relative complement of the class S exists as a subset of the base set. (Contributed by RP, 26-Jun-2021)

	Ref	Expression
Hypotheses	neicvgbex.d	|- D = ( P ` B )
	neicvgbex.h	|- H = ( F o. ( D o. G ) )
	neicvgbex.r	|- ( ph -> N H M )
Assertion	neicvgrcomplex	|- ( ph -> ( B \ S ) e. ~P B )

Proof

Step	Hyp	Ref	Expression
1		neicvgbex.d	|- D = ( P ` B )
2		neicvgbex.h	|- H = ( F o. ( D o. G ) )
3		neicvgbex.r	|- ( ph -> N H M )
4	1 2 3	neicvgbex	|- ( ph -> B e. _V )
5		difssd	|- ( ph -> ( B \ S ) C_ B )
6	4 5	sselpwd	|- ( ph -> ( B \ S ) e. ~P B )