Theorem psssstr 3376

Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996.)

Assertion

Ref	Expression
psssstr	⊢ ((A ⊊ B ∧ B ⊆ C) → A ⊊ C)

Proof of Theorem psssstr

Step	Hyp	Ref	Expression
1		sspss 3369	. 2 ⊢ (B ⊆ C ↔ (B ⊊ C ∨ B = C))
2		psstr 3374	. . . . 5 ⊢ ((A ⊊ B ∧ B ⊊ C) → A ⊊ C)
3	2	ex 423	. . . 4 ⊢ (A ⊊ B → (B ⊊ C → A ⊊ C))
4		psseq2 3358	. . . . 5 ⊢ (B = C → (A ⊊ B ↔ A ⊊ C))
5	4	biimpcd 215	. . . 4 ⊢ (A ⊊ B → (B = C → A ⊊ C))
6	3, 5	jaod 369	. . 3 ⊢ (A ⊊ B → ((B ⊊ C ∨ B = C) → A ⊊ C))
7	6	imp 418	. 2 ⊢ ((A ⊊ B ∧ (B ⊊ C ∨ B = C)) → A ⊊ C)
8	1, 7	sylan2b 461	1 ⊢ ((A ⊊ B ∧ B ⊆ C) → A ⊊ C)

Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358   = 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:  psssstrd  3379