Theorem eldif

Description: Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994)

	Ref	Expression
Assertion	eldif	|- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )

Proof

Step	Hyp	Ref	Expression
1		elex	|- ( A e. ( B \ C ) -> A e. _V )
2		elex	|- ( A e. B -> A e. _V )
3	2	adantr	|- ( ( A e. B /\ -. A e. C ) -> A e. _V )
4		eleq1	|- ( x = A -> ( x e. B <-> A e. B ) )
5		eleq1	|- ( x = A -> ( x e. C <-> A e. C ) )
6	5	notbid	|- ( x = A -> ( -. x e. C <-> -. A e. C ) )
7	4 6	anbi12d	|- ( x = A -> ( ( x e. B /\ -. x e. C ) <-> ( A e. B /\ -. A e. C ) ) )
8		df-dif	|- ( B \ C ) = { x | ( x e. B /\ -. x e. C ) }
9	7 8	elab2g	|- ( A e. _V -> ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) ) )
10	1 3 9	pm5.21nii	|- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )