Trichotomy is Antireflexive

Theorem
Let $\RR$ be a trichotomy.

Then $\RR$ is an antireflexive relation.

Proof
Let $\RR$ be a trichotomy on a set $S$.

Let $x \in S$.

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


 * $a \mathrel \RR b$
 * $a = b$
 * $b \mathrel \RR a$

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

Hence the result by definition of antireflexive relation.