Theorem sylan9ss 3196

Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Hypotheses

Ref	Expression
sylan9ss.1	|- ( ph -> A C_ B )
sylan9ss.2	|- ( ps -> B C_ C )

Assertion

Ref	Expression
sylan9ss	|- ( ( ph /\ ps ) -> A C_ C )

Proof of Theorem sylan9ss

Step	Hyp	Ref	Expression
1		sylan9ss.1	. 2 |- ( ph -> A C_ B )
2		sylan9ss.2	. 2 |- ( ps -> B C_ C )
3		sstr 3191	. 2 |- ( ( A C_ B /\ B C_ C ) -> A C_ C )
4	1, 2, 3	syl2an 289	1 |- ( ( ph /\ ps ) -> A C_ C )

Syntax hints:   -> wi 4   /\ wa 104   C_ wss 3157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170

This theorem is referenced by:  sylan9ssr  3197  unss12  3335  ss2in  3391  relrelss  5196  funssxp  5427