Relation Reflexivity

Theorem
Every relation has exactly one of these properties: it is either reflexive, antireflexive or non-reflexive.

The null relation is antireflexive.

Proof
A reflexive relation can not be antireflexive, and vice versa:

$$\left({x, x}\right) \in \mathcal{R} \iff \lnot \left({\left({x, x}\right) \notin \mathcal{R}}\right)$$

By the definition of non-reflexive, a reflexive relation can also not be non-reflexive.

So a reflexive relation is neither antireflexive nor non-reflexive.

An antireflexive relation can be neither reflexive (see above) nor non-reflexive (by the definition of non-reflexive).

By its own definition, if a relation is neither reflexive nor antireflexive, then it is non-reflexive.

Q.E.D.

The null relation is antireflexive:

This follows directly from the definition:

$$\mathcal{R} = \varnothing \Longrightarrow \forall x \in S: \left({x, x}\right) \notin \mathcal{R}$$