Theorem ntrneircomplex

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	ntrnei.o	|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
	ntrnei.f	|- F = ( ~P B O B )
	ntrnei.r	|- ( ph -> I F N )
Assertion	ntrneircomplex	|- ( ph -> ( B \ S ) e. ~P B )

Proof

Step	Hyp	Ref	Expression
1		ntrnei.o	|- O = ( i e. _V , j e. _V |-> ( k e. ( ~P j ^m i ) |-> ( l e. j |-> { m e. i | l e. ( k ` m ) } ) ) )
2		ntrnei.f	|- F = ( ~P B O B )
3		ntrnei.r	|- ( ph -> I F N )
4	1 2 3	ntrneibex	|- ( ph -> B e. _V )
5		difssd	|- ( ph -> ( B \ S ) C_ B )
6	4 5	sselpwd	|- ( ph -> ( B \ S ) e. ~P B )