Theorem difsnb

Description: ( B \ { A } ) equals B if and only if A is not a member of B . Generalization of difsn . (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Assertion	difsnb	|- ( -. A e. B <-> ( B \ { A } ) = B )

Proof

Step	Hyp	Ref	Expression
1		difsn	|- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd	|- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1	|- ( ( A e. B /\ -. A e. ( B \ { A } ) ) -> B =/= ( B \ { A } ) )
4	2 3	mpdan	|- ( A e. B -> B =/= ( B \ { A } ) )
5	4	necomd	|- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi	|- ( ( B \ { A } ) = B -> -. A e. B )
7	1 6	impbii	|- ( -. A e. B <-> ( B \ { A } ) = B )