Power 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 a inclusion lattice of power set of $X$.

Then $\mathcal P\left({X}\right)$ is a filter on $L$.

Filtered
By Set is Element of its Power Set:
 * $X \in \mathcal P\left({X}\right)$

Thus by definition:
 * $\mathcal P\left({X}\right)$ is a non-empty set.

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

By Intersection is Subset:
 * $x \cap y \subseteq x$ and $x \cap y \subseteq y$

By Subset Relation is Transitive:
 * $x \cap y \in \mathcal P\left({X}\right)$

Thus
 * $\exists z \in \mathcal P\left({X}\right): z \subseteq x \land z \subseteq y$

Upper
Thus we have:
 * $\forall x, y \in \mathcal P\left({X}\right): x \subseteq y \implies y \in \mathcal P\left({X}\right)$

Thus by definition of filter in ordered set:
 * $\mathcal P\left({X}\right)$ is a filter on $L$.