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.

$\blacksquare$

Also see

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

Sources

• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations: Exercise $4$