Theorem difsnb 3671

Description: ( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3665. (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 3665	. 2 |- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd 3662	. . . . 5 |- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1 2399	. . . . 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 2395	. . 3 |- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi 2364	. 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 1332   e. wcel 1481   =/= wne 2309   \ cdif 3073   {csn 3532

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-dif 3078  df-sn 3538

This theorem is referenced by: (None)