Equivalence of Definitions of Strict Ordering

Theorem
Let $S$ be a set.

Let $\mathcal R$ be a relation on $S$.

Then the following definitions for $\mathcal R$ to be a strict ordering are equivalent:

Proof
Let $\mathcal R$ be transitive.

Then by Transitive Relation is Antireflexive iff Asymmetric it follows directly that:


 * $(1): \quad$ If $\mathcal R$ is antireflexive then it is asymmetric
 * $(2): \quad$ If $\mathcal R$ is asymmetric then it is antireflexive.