Trichotomy Law (Ordering)

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

Then $$\preceq$$ is a total ordering iff:
 * $$\forall a, b \in S: a \prec b \lor a = b \lor a \succ b$$

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

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

Proof
$$ $$ $$ $$

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