Reflexive Reduction is Antireflexive

Theorem
Let $\RR$ be a relation on a set $S$.

Let $\RR^\ne$, be the reflexive reduction of $\RR$.

Then $\RR^\ne$ is antireflexive.

Proof
By the definition of reflexive reduction:


 * $\RR^\ne = \RR \setminus \Delta_S$

where $\Delta_S$ denotes the diagonal relation on $S$.

By Set Difference Intersection with Second Set is Empty Set:


 * $\paren {\RR \setminus \Delta_S} \cap \Delta_S = \O$

Hence by Relation is Antireflexive iff Disjoint from Diagonal Relation, $\RR^\ne$ is antireflexive.