Theorem sylan9ss 3214

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 3209	. 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 3174

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187

This theorem is referenced by:  sylan9ssr  3215  unss12  3353  ss2in  3409  relrelss  5228  funssxp  5465