Trichotomy Law (Ordering)

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Then $\preceq$ is a total ordering iff:
 * $\forall a, b \in S: \left({a \prec b}\right) \lor \left({a = b}\right) \lor \left({a \succ b}\right)$

That is, every element either strictly precedes, is the same as, or strictly succeeds, every other element.

In other words, iff $\prec$ is a trichotomy.

Comment
Simple and obvious, but as important and far-reaching as the Law of the Excluded Middle.