Antireflexive and Transitive Relation is Antisymmetric

Theorem
Let $\RR \subseteq S \times S$ be a relation which is not null.

Let $\RR$ be antireflexive and transitive.

Then $\RR$ is also antisymmetric.

Proof
Let $\RR \subseteq S \times S$ be antireflexive and transitive.

From Antireflexive and Transitive Relation is Asymmetric it follows that $\RR$ is asymmetric.

The result follows from Asymmetric Relation is Antisymmetric.

Also see
If $\RR = \O$ then Null Relation is Antireflexive, Symmetric and Transitive applies instead.