ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sylan9ss Unicode version

Theorem sylan9ss 3168
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 3163 . 2  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
41, 2, 3syl2an 289 1  |-  ( (
ph  /\  ps )  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    C_ wss 3129
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142
This theorem is referenced by:  sylan9ssr  3169  unss12  3307  ss2in  3363  relrelss  5150  funssxp  5380
  Copyright terms: Public domain W3C validator