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

Theorem sopo 4070
Description: A strict linear order is a strict partial order. (Contributed by NM, 28-Mar-1997.)
Assertion
Ref Expression
sopo (𝑅 Or 𝐴𝑅 Po 𝐴)

Proof of Theorem sopo
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iso 4054 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
21simplbi 268 1 (𝑅 Or 𝐴𝑅 Po 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662  wral 2349   class class class wbr 3787   Po wpo 4051   Or wor 4052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104
This theorem depends on definitions:  df-bi 115  df-iso 4054
This theorem is referenced by:  sonr  4074  sotr  4075  so2nr  4078  so3nr  4079  sosng  4433
  Copyright terms: Public domain W3C validator