Theorem difsnpss 4111

Description: ( B \ { A } ) is a proper subclass of B if and only if A is a member of B. (Contributed by David Moews, 1-May-2017.)

Assertion

Ref	Expression
difsnpss	|- ( A e. B <-> ( B \ { A } ) C. B )

Proof of Theorem difsnpss

Step	Hyp	Ref	Expression
1		notnot 291	. 2 |- ( A e. B <-> -. -. A e. B )
2		difss 3578	. . . 4 |- ( B \ { A } ) C_ B
3	2	biantrur 506	. . 3 |- ( ( B \ { A } ) =/= B <-> ( ( B \ { A } ) C_ B /\ ( B \ { A } ) =/= B ) )
4		difsnb 4110	. . . 4 |- ( -. A e. B <-> ( B \ { A } ) = B )
5	4	necon3bbii 2707	. . 3 |- ( -. -. A e. B <-> ( B \ { A } ) =/= B )
6		df-pss 3439	. . 3 |- ( ( B \ { A } ) C. B <-> ( ( B \ { A } ) C_ B /\ ( B \ { A } ) =/= B ) )
7	3, 5, 6	3bitr4i 277	. 2 |- ( -. -. A e. B <-> ( B \ { A } ) C. B )
8	1, 7	bitri 249	1 |- ( A e. B <-> ( B \ { A } ) C. B )

Syntax hints:   -. wn 3   <-> wb 184   /\ wa 369   e. wcel 1758   =/= wne 2642   \ cdif 3420   C_ wss 3423   C. wpss 3424   {csn 3972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-v 3067  df-dif 3426  df-in 3430  df-ss 3437  df-pss 3439  df-sn 3973

This theorem is referenced by:  marypha1lem  7781  infpss  8484  ominf4  8579  mrieqv2d  14676