Lower Closure of Element is Closed under Directed Suprema

Theorem
Let $L = \struct {S, \preceq}$ be an up-complete ordered set.

Let $x \in S$.

Then $x^\preceq$ is closed under directed suprema,

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

Proof
Let $D$ be a directed subset of $S$ such that
 * $D \subseteq x^\preceq$

By Lower Closure of Element is Ideal:
 * $x^\preceq$ is directed.

By definition of up-complete:
 * $D$ and $x^\preceq$ admit suprema.

By Supremum of Subset:
 * $\sup D \preceq \map \sup {x^\preceq}$

By Supremum of Lower Closure of Element:
 * $\sup D \preceq x$

Thus by definition of lower closure of element:
 * $\sup D \in x^\preceq$