Theorem difsnb 3853

Description: ( B \ { A } ) equals B if and only if A is not a member of B. Generalization of difsn 3847. (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 3847	. 2 |- ( -. A e. B -> ( B \ { A } ) = B )
2		neldifsnd 3840	. . . . 5 |- ( A e. B -> -. A e. ( B \ { A } ) )
3		nelne1 2510	. . . . 5 |- ( ( A e. B /\ -. A e. ( B \ { A } ) ) -> B =/= ( B \ { A } ) )
4	2, 3	mpdan 425	. . . 4 |- ( A e. B -> B =/= ( B \ { A } ) )
5	4	necomd 2506	. . 3 |- ( A e. B -> ( B \ { A } ) =/= B )
6	5	necon2bi 2475	. 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 1402   e. wcel 2209   =/= wne 2420   \ cdif 3217   {csn 3705

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-sn 3711

This theorem is referenced by: (None)