Dual Ordering is Ordering

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $\succeq$ denote the dual ordering of $\preceq$.

Then $\succeq$ is an ordering on $S$.

Proof
By definition, $\succeq$ is the inverse relation to $\preceq$.

By Inverse Relation Properties, if a relation is reflexive, transitive and/or antisymmetric, then so is its inverse.

The result follows.