Singleton is Directed and Filtered Subset

Theorem
Let $\struct {S, \precsim}$ be a preordered set.

Let $x$ be an element of $S$.

Then $\set x$ is directed and filtered subset of $S$.

Proof
Let $y, z \in \set x$.

By definition of singleton:
 * $ y = x \land z = x$

By definition of reflexivity:
 * $y \precsim x \land z \precsim x$

Thus:
 * $\exists h \in \set x: y \precsim h \land z \precsim h$

Thus by definition:
 * $\set x$ is a directed subset of $S$.

$\set x$ is a filtered subset of $S$ by mutatis mutandis.