Restriction of Antireflexive Relation is Antireflexive

Theorem
Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be an antireflexive relation on $S$.

Let $T \subseteq S$ be a subset of $S$.

Let $\mathcal R \restriction_T \ \subseteq T \times T$ be the restriction of $\mathcal R$ to $T$.

Then $\mathcal R \restriction_T$ is an antireflexive relation on $T$.

Proof
Suppose $\mathcal R$ is antireflexive on $S$.

Then:
 * $\forall x \in S: \left({x, x}\right) \notin \mathcal R$

So:
 * $\forall x \in T: \left({x, x}\right) \notin \mathcal R \restriction_T$

Thus $\mathcal R \restriction_T$ is antireflexive on $T$.

Also see

 * Properties of Restriction of Relation‎ for other similar properties of the restriction of a relation.