Transitive Relation is Antireflexive iff Asymmetric

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

Let $\RR$ be transitive.

Then $\RR$ is antireflexive $\RR$ is asymmetric.

Necessary Condition
Let $\RR \subseteq S \times S$ be antireflexive.

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

Sufficient Condition
Let $\RR$ be asymmetric.

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

Also see

 * Null Relation is Antireflexive, Symmetric and Transitive for the case where $\RR = \O$.