Theorem sylan9ss 3155

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 3150	. 2 |- ( ( A C_ B /\ B C_ C ) -> A C_ C )
4	1, 2, 3	syl2an 287	1 |- ( ( ph /\ ps ) -> A C_ C )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  sylan9ssr  3156  unss12  3294  ss2in  3350  relrelss  5130  funssxp  5357