Lower Closure of Directed Subset is Ideal

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

Let $D$ be a directed subset of $S$.

Then
 * $D^\preceq$ is an ideal in $\mathscr S$

where $D^\preceq$ denotes the lower closure of $D$.

Proof
By Directed iff Lower Closure Directed:
 * $D^\preceq$ is directed.

By Lower Closure is Lower Set:
 * $D^\preceq$ is a lower set.

Thus by definition
 * $D^\preceq$ is an ideal in $\mathscr S$