Theorem difsnb 3506

Description: ( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3501. (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 3501	. 2 |- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd 3498	. . . . 5 |- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1 2295	. . . . 5 |- ( ( A e. B /\ -. A e. ( B \ { A } ) ) -> B =/= ( B \ { A } ) )
4	2, 3	mpdan 398	. . . 4 |- ( A e. B -> B =/= ( B \ { A } ) )
5	4	necomd 2291	. . 3 |- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi 2260	. 2 |- ( ( B \ { A } ) = B -> -. A e. B )
7	1, 6	impbii 117	1 |- ( -. A e. B <-> ( B \ { A } ) = B )

Syntax hints:   -. wn 3   <-> wb 98   = wceq 1243   e. wcel 1393   =/= wne 2204   \ cdif 2914   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-sn 3381

This theorem is referenced by:  difsnpssim  3507