Theorem difsnb 4337

Description: ( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 4328. (Contributed by David Moews, 1-May-2017.)

Assertion

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

Proof of Theorem difsnb

Step	Hyp	Ref	Expression
1		difsn 4328	. 2 |- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd 4322	. . . . 5 |- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1 2890	. . . . 5 |- ( ( A e. B /\ -. A e. ( B \ { A } ) ) -> B =/= ( B \ { A } ) )
4	2, 3	mpdan 702	. . . 4 |- ( A e. B -> B =/= ( B \ { A } ) )
5	4	necomd 2849	. . 3 |- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi 2824	. 2 |- ( ( B \ { A } ) = B -> -. A e. B )
7	1, 6	impbii 199	1 |- ( -. A e. B <-> ( B \ { A } ) = B )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-sn 4178

This theorem is referenced by:  difsnpss  4338  incexclem  14568  mrieqv2d  16299  mreexmrid  16303  mreexexlem2d  16305  mreexexlem4d  16307  acsfiindd  17177