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Theorem sopo 4113
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 4097 . 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 2355   class class class wbr 3820   Po wpo 4094   Or wor 4095
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 4097
This theorem is referenced by:  sonr  4117  sotr  4118  so2nr  4121  so3nr  4122  sosng  4478  fimaxq  10124
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