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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
StepHypRef 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 )
41, 2, 3syl2an 283 1  |-  ( (
ph  /\  ps )  ->  A  C_  C )
Colors of variables: wff set class
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
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