Singleton of Set is Filter in Lattice of Power Set

Theorem
Let $X$ be a set.

Let $L = \struct {\powerset X, \cup, \cap, \subseteq}$ be an inclusion lattice of power set of $X$.

Then $\set X$ is a filter on $L$.

Proof
By Singleton is Directed and Filtered Subset:
 * $\set X$ is filtered.

We will prove that
 * $\set X$ is an upper set.

Let $x \in \set X$, $y \in \powerset X$ such that:
 * $x \subseteq y$

By definition of singleton:
 * $x = X$

By definition of power set:
 * $y \subseteq X$

By definition of set equality:
 * $y = X$

Thus:
 * $y \in \set X$

Thus by definition of filter in ordered set:
 * $\set X$ is a filter.