Trichotomy Law (Ordering)

Theorem
Let $$\left({S; \le}\right)$$ be a poset.

Then $$\le$$ is a total ordering iff:

$$\forall a, b \in S: a  b$$

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

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