Theorem difsnpss 4338

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		notnotb 304	. 2 |- ( A e. B <-> -. -. A e. B )
2		difss 3737	. . . 4 |- ( B \ { A } ) C_ B
3	2	biantrur 527	. . 3 |- ( ( B \ { A } ) =/= B <-> ( ( B \ { A } ) C_ B /\ ( B \ { A } ) =/= B ) )
4		difsnb 4337	. . . 4 |- ( -. A e. B <-> ( B \ { A } ) = B )
5	4	necon3bbii 2841	. . 3 |- ( -. -. A e. B <-> ( B \ { A } ) =/= B )
6		df-pss 3590	. . 3 |- ( ( B \ { A } ) C. B <-> ( ( B \ { A } ) C_ B /\ ( B \ { A } ) =/= B ) )
7	3, 5, 6	3bitr4i 292	. 2 |- ( -. -. A e. B <-> ( B \ { A } ) C. B )
8	1, 7	bitri 264	1 |- ( A e. B <-> ( B \ { A } ) C. B )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   C. wpss 3575   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pss 3590  df-sn 4178

This theorem is referenced by:  marypha1lem  8339  infpss  9039  ominf4  9134  mrieqv2d  16299