Definition 1.4.15.  Let S be a set, ⪯ be a relation on S. We define:
⪯ is a partial order  :⇔  ⪯ is reflexive and ⪯ is antisymmetric and ⪯ is transitive  ⇔  ⪯ is a preorder and ⪯ is antisymmetric
Equivalence.  No proof.
References.