Antireflexive and Transitive Relation is Antisymmetric

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

Let $\mathcal R$ be antireflexive and transitive.

Then $\mathcal R$ is also antisymmetric.

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

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

The result follows from Asymmetric Relation is Antisymmetric.

Also see
If $\mathcal R = \varnothing$ then Null Relation is Antireflexive, Symmetric and Transitive applies instead.