Theorem sspss 3368

Description: Subclass in terms of proper subclass. (Contributed by NM, 25-Feb-1996.)

Assertion

Ref	Expression
sspss	|- (A C_ B <-> (A C. B \/ A = B))

Proof of Theorem sspss

Step	Hyp	Ref	Expression
1		dfpss2 3354	. . . . 5 |- (A C. B <-> (A C_ B /\ -. A = B))
2	1	simplbi2 608	. . . 4 |- (A C_ B -> (-. A = B -> A C. B))
3	2	con1d 116	. . 3 |- (A C_ B -> (-. A C. B -> A = B))
4	3	orrd 367	. 2 |- (A C_ B -> (A C. B \/ A = B))
5		pssss 3364	. . 3 |- (A C. B -> A C_ B)
6		eqimss 3323	. . 3 |- (A = B -> A C_ B)
7	5, 6	jaoi 368	. 2 |- ((A C. B \/ A = B) -> A C_ B)
8	4, 7	impbii 180	1 |- (A C_ B <-> (A C. B \/ A = B))

Syntax hints:  -. wn 3   <-> wb 176   \/ wo 357   = wceq 1642   C_ wss 3257   C. wpss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pss 3261

This theorem is referenced by:  sspsstri  3371  sspsstr  3374  psssstr  3375  ssfin  4470