Theorem syl6ss 3285

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 3283	1 ⊢ (φ → A ⊆ C)

Syntax hints:   → wi 4   ⊆ 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:  difss2  3396  rintn0  4057  pw1equn  4332  pw1eqadj  4333  peano5  4410  spfininduct  4541  vfinncvntsp  4550  vfinspsslem1  4551  vfinncsp  4555  ssxpb  5056  funssxp  5234  dff2  5420  dff3  5421  dff4  5422  clos1induct  5881  dmfrec  6317  frecsuc  6323