MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sylan9ss Unicode version

Theorem sylan9ss 3205
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 3200 . 2  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
41, 2, 3syl2an 463 1  |-  ( (
ph  /\  ps )  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    C_ wss 3165
This theorem is referenced by:  sylan9ssr  3206  psstr  3293  unss12  3360  ss2in  3409  relrelss  5212  funssxp  5418  axdc3lem  8092  tskuni  8421  tsmsxp  17853  shslubi  21980  chlej12i  22070  rtrclreclem.min  24059  fnetr  26389  pcl0bN  30734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179
  Copyright terms: Public domain W3C validator