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

Theorem sstrd 3010
Description: Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.)
Hypotheses
Ref Expression
sstrd.1  |-  ( ph  ->  A  C_  B )
sstrd.2  |-  ( ph  ->  B  C_  C )
Assertion
Ref Expression
sstrd  |-  ( ph  ->  A  C_  C )

Proof of Theorem sstrd
StepHypRef Expression
1 sstrd.1 . 2  |-  ( ph  ->  A  C_  B )
2 sstrd.2 . 2  |-  ( ph  ->  B  C_  C )
3 sstr 3008 . 2  |-  ( ( A  C_  B  /\  B  C_  C )  ->  A  C_  C )
41, 2, 3syl2anc 403 1  |-  ( ph  ->  A  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    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:  syl5ss  3011  syl6ss  3012  ssdif2d  3112  tfisi  4330  funss  4944  fssxp  5083  fvmptssdm  5281  suppssfv  5733  suppssov1  5734  tposss  5889  tfrlem1  5951  tfrlemibfn  5971  tfr1onlembfn  5987  tfr1onlemubacc  5989  tfr1onlemres  5992  tfrcllembfn  6000  tfrcllemubacc  6002  tfrcllemres  6005  ecinxp  6240  undiffi  6433
  Copyright terms: Public domain W3C validator