Dual Ordering is Ordering

Theorem
If $\preceq$ is a partial ordering on $S$, then so is its inverse $\preceq^{-1}$.

The inverse of an ordering is usually denoted by reversing its symbol, thus $\preceq^{-1}$ is written $\succeq$.

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

The result follows.