Theorem difsnpss 4086

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 292	. 2 |- ( A e. B <-> -. -. A e. B )
2		difss 3535	. . . 4 |- ( B \ { A } ) C_ B
3	2	biantrur 508	. . 3 |- ( ( B \ { A } ) =/= B <-> ( ( B \ { A } ) C_ B /\ ( B \ { A } ) =/= B ) )
4		difsnb 4085	. . . 4 |- ( -. A e. B <-> ( B \ { A } ) = B )
5	4	necon3bbii 2648	. . 3 |- ( -. -. A e. B <-> ( B \ { A } ) =/= B )
6		df-pss 3395	. . 3 |- ( ( B \ { A } ) C. B <-> ( ( B \ { A } ) C_ B /\ ( B \ { A } ) =/= B ) )
7	3, 5, 6	3bitr4i 280	. 2 |- ( -. -. A e. B <-> ( B \ { A } ) C. B )
8	1, 7	bitri 252	1 |- ( A e. B <-> ( B \ { A } ) C. B )

Syntax hints:   -. wn 3   <-> wb 187   /\ wa 370   e. wcel 1872   =/= wne 2599   \ cdif 3376   C_ wss 3379   C. wpss 3380   {csn 3941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-v 3024  df-dif 3382  df-in 3386  df-ss 3393  df-pss 3395  df-sn 3942

This theorem is referenced by:  marypha1lem  7900  infpss  8598  ominf4  8693  mrieqv2d  15488