Theorem difsnb 3663

Description: ( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3657. (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 3657	. 2 |- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd 3654	. . . . 5 |- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1 2398	. . . . 5 |- ( ( A e. B /\ -. A e. ( B \ { A } ) ) -> B =/= ( B \ { A } ) )
4	2, 3	mpdan 417	. . . 4 |- ( A e. B -> B =/= ( B \ { A } ) )
5	4	necomd 2394	. . 3 |- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi 2363	. 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 1331   e. wcel 1480   =/= wne 2308   \ cdif 3068   {csn 3527

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-dif 3073  df-sn 3533

This theorem is referenced by: (None)