Theorem difsnb 3716

Description: ( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3710. (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 3710	. 2 |- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd 3707	. . . . 5 |- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1 2426	. . . . 5 |- ( ( A e. B /\ -. A e. ( B \ { A } ) ) -> B =/= ( B \ { A } ) )
4	2, 3	mpdan 418	. . . 4 |- ( A e. B -> B =/= ( B \ { A } ) )
5	4	necomd 2422	. . 3 |- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi 2391	. 2 |- ( ( B \ { A } ) = B -> -. A e. B )
7	1, 6	impbii 125	1 |- ( -. A e. B <-> ( B \ { A } ) = B )

Syntax hints:   -. wn 3   <-> wb 104   = wceq 1343   e. wcel 2136   =/= wne 2336   \ cdif 3113   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-dif 3118  df-sn 3582

This theorem is referenced by: (None)