Theorem difsnb 3723

Description: ( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3717. (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 3717	. 2 |- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd 3714	. . . . 5 |- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1 2430	. . . . 5 |- ( ( A e. B /\ -. A e. ( B \ { A } ) ) -> B =/= ( B \ { A } ) )
4	2, 3	mpdan 419	. . . 4 |- ( A e. B -> B =/= ( B \ { A } ) )
5	4	necomd 2426	. . 3 |- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi 2395	. 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 1348   e. wcel 2141   =/= wne 2340   \ cdif 3118   {csn 3583

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-v 2732  df-dif 3123  df-sn 3589

This theorem is referenced by: (None)