Theorem difsnb 3749

Description: ( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3743. (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 3743	. 2 |- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd 3737	. . . . 5 |- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1 2449	. . . . 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 2445	. . 3 |- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi 2414	. 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 1363   e. wcel 2159   =/= wne 2359   \ cdif 3140   {csn 3606

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-v 2753  df-dif 3145  df-sn 3612

This theorem is referenced by: (None)