Theorem difsnb 3580

Description: ( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3574. (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 3574	. 2 |- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd 3571	. . . . 5 |- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1 2345	. . . . 5 |- ( ( A e. B /\ -. A e. ( B \ { A } ) ) -> B =/= ( B \ { A } ) )
4	2, 3	mpdan 412	. . . 4 |- ( A e. B -> B =/= ( B \ { A } ) )
5	4	necomd 2341	. . 3 |- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi 2310	. 2 |- ( ( B \ { A } ) = B -> -. A e. B )
7	1, 6	impbii 124	1 |- ( -. A e. B <-> ( B \ { A } ) = B )

Syntax hints:   -. wn 3   <-> wb 103   = wceq 1289   e. wcel 1438   =/= wne 2255   \ cdif 2996   {csn 3446

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 3001  df-sn 3452

This theorem is referenced by: (None)