Filter is Ideal in Dual Ordered Set

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

Let $X$ be a subset of $S$.

Then
 * $X$ is filter in $P$


 * $X$ is ideal in $P^{-1}$

where $P^{-1} = \left({S, \succeq}\right)$ denotes the dual of $P$.

Proof
By Dual of Dual Ordering:
 * dual of $P^{-1}$ is $P$.

Hence by Ideal is Filter in Dual Ordered Set:
 * the result fallows.