Null Relation is Antireflexive, Symmetric and Transitive

Theorem
Let $$S$$ be a set which is not empty.

Let $$\mathcal R \subseteq S \times S$$ be the null relation.

Then $$\mathcal R$$ is antireflexive, symmetric and transitive.

If $$S = \varnothing$$ then Relation on Null Set is Equivalence applies.

Proof
From the definition of null relation, $$\mathcal R = \varnothing$$.

Antireflexivity
This follows directly from the definition:
 * $$\mathcal R = \varnothing \implies \forall x \in S: \left({x, x}\right) \notin \mathcal R$$

and so $$\mathcal R$$ is antireflexive.

Symmetry
It follows vacuously that:
 * $$\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \in \mathcal R$$

and so $$\mathcal R$$ is symmetric.

Transitivity
It follows vacuously that:
 * $$\left({x, y}\right), \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \in \mathcal R$$

and so $$\mathcal R$$ is transitive.