Reflexive Reduction of Ordering is Strict Ordering/Proof 2

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

Then $\mathcal R^\neq$, the reflexive reduction of $\mathcal R$, is a strict ordering on $S$.

Proof
Follows from Reflexive Reduction of Transitive Antisymmetric Relation is Strict Ordering.