Theorem sstri 3282

Description: Subclass transitivity inference. (Contributed by NM, 5-May-2000.)

Hypotheses

Ref	Expression
sstri.1	⊢ A ⊆ B
sstri.2	⊢ B ⊆ C

Assertion

Ref	Expression
sstri	⊢ A ⊆ C

Proof of Theorem sstri

Step	Hyp	Ref	Expression
1		sstri.1	. 2 ⊢ A ⊆ B
2		sstri.2	. 2 ⊢ B ⊆ C
3		sstr2 3280	. 2 ⊢ (A ⊆ B → (B ⊆ C → A ⊆ C))
4	1, 2, 3	mp2 17	1 ⊢ A ⊆ C

Syntax hints:   ⊆ 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:  snsstp1  3859  snsstp2  3860  uniintsn  3964  inxpk  4278  dfidk2  4314  ssrnres  5060  fvfullfunlem3  5864  sbthlem1  6204  spacssnc  6285