Ideal is Filter in Dual Ordered Set

Theorem
Let $P = \struct {S, \preceq}$ be an ordered set.

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

Then
 * $X$ is ideal in $P$


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

where $P^{-1} = \struct {S, \succeq}$ denotes the dual of $P$.

Sufficient Condition
Let $X$ be ideal in $P$.

By definition of ideal in ordered set:
 * $X$ is non-empty directed lower.

By definition of directed:
 * $\forall x, y \in X: \exists z \in X: x \preceq z \land y \preceq z$

Then
 * $\forall x, y \in X: \exists z \in X: z \succeq x \land z \succeq y$

By definition
 * $X$ is filtered in $P^{-1}$

By definition of lower section:
 * $\forall x \in X, y \in S: y \preceq x \implies y \in X$

Then
 * $\forall x \in X, y \in S: x \succeq y \implies y \in X$

By definition
 * $X$ is an upper section in $P^{-1}$.

Thus by definition of filter in ordered set
 * $X$ is a filter in $P^{-1}$.

Necessary Condition
Let $X$ be a filter in $P^{-1}$.

By definition of filter in ordered set
 * $X$ is non-empty, filtered in $P^{-1}$, and upper in $P^{-1}$.

By definition of filtered in $P^{-1}$:
 * $\forall x, y \in X: \exists z \in X: z \succeq x \land z \succeq y$

Then
 * $\forall x, y \in X: \exists z \in X: x \preceq z \land y \preceq z$

By definition:
 * $X$ is directed.

By definition of upper section in $P^{-1}$:
 * $\forall x \in X, y \in S: x \succeq y \implies y \in X$

Then:
 * $\forall x \in X, y \in S: y \preceq x \implies y \in X$

By definition:
 * $X$ is a lower section.

Thus by definition of ideal in ordered set:
 * $X$ is an ideal in $P$.