Trichotomy is Antireflexive

Theorem
Let $\mathcal R$ be a trichotomy.

Then $\mathcal R$ is an irreflexive relation.

Proof
Let $\mathcal R$ be a trichotomy on a set $S$.

Let $x \in S$.

By definition of a trichotomy, for all $a, b \in S$, either:


 * $a \mathop {\mathcal R} b$
 * $a = b$
 * $b \mathop {\mathcal R} a$

As $x = x$ it follows directly that $x \not < x$.

Hence the result by definition of irreflexive relation.