Singleton of Set is Filter in Lattice of Power Set

Theorem
Let $X$ be a set.

Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \subseteq}\right)$ be an inclusion lattice of power set of $X$.

Then $\left\{ {X}\right\}$ is a filter on $L$.

Proof
By Singleton is Directed and Filtered Subset:
 * $\left\{ {X}\right\}$ is filtered.

We will prove that
 * $\left\{ {X}\right\}$ is an upper set.

Let $x \in \left\{ {X}\right\}$, $y \in \mathcal P\left({X}\right)$ 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:
 * $y \in \left\{ {X}\right\}$

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