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Theorem sylan9ss 3361
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 3356 . 2  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
41, 2, 3syl2an 464 1  |-  ( (
ph  /\  ps )  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    C_ wss 3320
This theorem is referenced by:  sylan9ssr  3362  psstr  3451  unss12  3519  ss2in  3568  relrelss  5393  funssxp  5604  axdc3lem  8330  tskuni  8658  tsmsxp  18184  shslubi  22887  chlej12i  22977  insiga  24520  rtrclreclem.min  25147  fnetr  26366  pcl0bN  30720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-in 3327  df-ss 3334
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