Trichotomy Law (Ordering)

Theorem
Let $\struct {S, \preceq}$ be an ordered set.

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

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

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