Theorem difsnb 3735

Description: ( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3729. (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 3729	. 2 |- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd 3723	. . . . 5 |- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1 2437	. . . . 5 |- ( ( A e. B /\ -. A e. ( B \ { A } ) ) -> B =/= ( B \ { A } ) )
4	2, 3	mpdan 421	. . . 4 |- ( A e. B -> B =/= ( B \ { A } ) )
5	4	necomd 2433	. . 3 |- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi 2402	. 2 |- ( ( B \ { A } ) = B -> -. A e. B )
7	1, 6	impbii 126	1 |- ( -. A e. B <-> ( B \ { A } ) = B )

Syntax hints:   -. wn 3   <-> wb 105   = wceq 1353   e. wcel 2148   =/= wne 2347   \ cdif 3126   {csn 3592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2739  df-dif 3131  df-sn 3598

This theorem is referenced by: (None)