Total Ordering is Total Relation

Theorem
Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be a total ordering.

Then $\mathcal R$ is a total relation.

Proof
By definition of total ordering:


 * $\mathcal R$ is a reflexive relation on the strength of being an ordering
 * $\mathcal R$ is a connected relation on the strength of being a total ordering.