Theorem sspss 3369

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

Assertion

Ref	Expression
sspss	⊢ (A ⊆ B ↔ (A ⊊ B ∨ A = B))

Proof of Theorem sspss

Step	Hyp	Ref	Expression
1		dfpss2 3355	. . . . 5 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ ¬ A = B))
2	1	simplbi2 608	. . . 4 ⊢ (A ⊆ B → (¬ A = B → A ⊊ B))
3	2	con1d 116	. . 3 ⊢ (A ⊆ B → (¬ A ⊊ B → A = B))
4	3	orrd 367	. 2 ⊢ (A ⊆ B → (A ⊊ B ∨ A = B))
5		pssss 3365	. . 3 ⊢ (A ⊊ B → A ⊆ B)
6		eqimss 3324	. . 3 ⊢ (A = B → A ⊆ B)
7	5, 6	jaoi 368	. 2 ⊢ ((A ⊊ B ∨ A = B) → A ⊆ B)
8	4, 7	impbii 180	1 ⊢ (A ⊆ B ↔ (A ⊊ B ∨ A = B))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   = wceq 1642   ⊆ wss 3258   ⊊ wpss 3259

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 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pss 3262

This theorem is referenced by:  sspsstri  3372  sspsstr  3375  psssstr  3376  ssfin  4471