Theorem sylan9ss 3013

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 3008	. 2 |- ( ( A C_ B /\ B C_ C ) -> A C_ C )
4	1, 2, 3	syl2an 283	1 |- ( ( ph /\ ps ) -> A C_ C )

Syntax hints:   -> wi 4   /\ wa 102   C_ wss 2974

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-in 2980  df-ss 2987

This theorem is referenced by:  sylan9ssr  3014  unss12  3145  ss2in  3200  relrelss  4874  funssxp  5091