Dual Ordering is Ordering

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

The inverse of an ordering is usually denoted by reversing its symbol, thus:

$$\le^{-1}$$ is written $$\ge$$

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

The result follows.