Theorem syl6ss 3284

Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
syl6ss.1	⊢ (φ → A ⊆ B)
syl6ss.2	⊢ B ⊆ C

Assertion

Ref	Expression
syl6ss	⊢ (φ → A ⊆ C)

Proof of Theorem syl6ss

Step	Hyp	Ref	Expression
1		syl6ss.1	. 2 ⊢ (φ → A ⊆ B)
2		syl6ss.2	. . 3 ⊢ B ⊆ C
3	2	a1i 10	. 2 ⊢ (φ → B ⊆ C)
4	1, 3	sstrd 3282	1 ⊢ (φ → A ⊆ C)

Syntax hints:   → wi 4   ⊆ wss 3257

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259

This theorem is referenced by:  difss2  3395  rintn0  4056  pw1equn  4331  pw1eqadj  4332  peano5  4409  spfininduct  4540  vfinncvntsp  4549  vfinspsslem1  4550  vfinncsp  4554  ssxpb  5055  funssxp  5233  dff2  5419  dff3  5420  dff4  5421  clos1induct  5880  dmfrec  6316  frecsuc  6322