Theorem sstr 3281

Description: Transitivity of subclasses. Theorem 6 of [Suppes] p. 23. (Contributed by NM, 5-Sep-2003.)

Assertion

Ref	Expression
sstr	⊢ ((A ⊆ B ∧ B ⊆ C) → A ⊆ C)

Proof of Theorem sstr

Step	Hyp	Ref	Expression
1		sstr2 3280	. 2 ⊢ (A ⊆ B → (B ⊆ C → A ⊆ C))
2	1	imp 418	1 ⊢ ((A ⊆ B ∧ B ⊆ C) → A ⊆ C)

Syntax hints:   → wi 4   ∧ wa 358   ⊆ wss 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  sstrd  3283  sylan9ss  3286  ssdifss  3398  uneqin  3507  sspw1  4336  ssrnres  5060  fco  5232  fssres  5239  ssetpov  5945  sbthlem1  6204