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

Theorem ispod 4032
Description: Sufficient conditions for a partial order. (Contributed by NM, 9-Jul-2014.)
Hypotheses
Ref Expression
ispod.1  R
ispod.2  R  R  R
Assertion
Ref Expression
ispod  R  Po
Distinct variable groups:   ,,,   , R,,   ,,,

Proof of Theorem ispod
StepHypRef Expression
1 ispod.1 . . . . 5  R
213ad2antr1 1068 . . . 4  R
3 ispod.2 . . . 4  R  R  R
42, 3jca 290 . . 3  R  R  R  R
54ralrimivvva 2396 . 2  R  R  R  R
6 df-po 4024 . 2  R  Po  R  R  R  R
75, 6sylibr 137 1  R  Po
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   w3a 884   wcel 1390  wral 2300   class class class wbr 3755    Po wpo 4022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-ral 2305  df-po 4024
This theorem is referenced by:  swopo  4034  pofun  4040  ltsopi  6304  ltsonq  6382  ltpopr  6569  ltposr  6691  ltso  6893  xrltso  8487
  Copyright terms: Public domain W3C validator