Theorem sspsstr 3374

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

Assertion

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

Proof of Theorem sspsstr

Step	Hyp	Ref	Expression
1		sspss 3368	. 2 ⊢ (A ⊆ B ↔ (A ⊊ B ∨ A = B))
2		psstr 3373	. . . . 5 ⊢ ((A ⊊ B ∧ B ⊊ C) → A ⊊ C)
3	2	ex 423	. . . 4 ⊢ (A ⊊ B → (B ⊊ C → A ⊊ C))
4		psseq1 3356	. . . . 5 ⊢ (A = B → (A ⊊ C ↔ B ⊊ C))
5	4	biimprd 214	. . . 4 ⊢ (A = B → (B ⊊ C → A ⊊ C))
6	3, 5	jaoi 368	. . 3 ⊢ ((A ⊊ B ∨ A = B) → (B ⊊ C → A ⊊ C))
7	6	imp 418	. 2 ⊢ (((A ⊊ B ∨ A = B) ∧ B ⊊ C) → A ⊊ C)
8	1, 7	sylanb 458	1 ⊢ ((A ⊆ B ∧ B ⊊ C) → A ⊊ C)

Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358   = wceq 1642   ⊆ wss 3257   ⊊ wpss 3258

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  sspsstrd  3377