Reflexive Reduction of Ordering is Strict Ordering/Proof 2

Theorem
Let $\mathcal R$ be an ordering on a set $S$.

Let $\mathcal R^\ne$ be the reflexive reduction of $\mathcal R$.

Then $\mathcal R^\ne$ is a strict ordering on $S$.

Proof
By definition, an ordering is both reflexive and transitive.

The result then follows from Reflexive Reduction of Transitive Antisymmetric Relation is Strict Ordering.