Theorem difsnb 3821

Description: ( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3815. (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 3815	. 2 |- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd 3808	. . . . 5 |- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1 2493	. . . . 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 2489	. . 3 |- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi 2458	. 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 1398   e. wcel 2202   =/= wne 2403   \ cdif 3198   {csn 3673

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-dif 3203  df-sn 3679

This theorem is referenced by: (None)