Transitive Relation is Antireflexive iff Asymmetric

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

Let $\mathcal R$ be transitive.

Then $\mathcal R$ is antireflexive iff $\mathcal R$ is asymmetric.

Necessary Condition
Let $\mathcal R \subseteq S \times S$ be antireflexive.

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

Sufficient Condition
Let $\mathcal R$ be asymmetric.

Then from Asymmetric Relation is Antireflexive it follows directly that $\mathcal R$ is antireflexive.

Also see

 * Null Relation is Antireflexive, Symmetric and Transitive for the case where $\mathcal R = \varnothing$.