Power Set is Filter in Lattice of Power Set

Theorem
Let $X$ be a set.

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

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

Filtered
By Set is Element of its Power Set:
 * $X \in \powerset X$

Thus by definition:
 * $\powerset X$ is a non-empty set.

Let $x, y \in \powerset X$.

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

By Subset Relation is Transitive:
 * $x \cap y \in \powerset X$

Thus
 * $\exists z \in \powerset X: z \subseteq x \land z \subseteq y$

Upper
Thus we have:
 * $\forall x, y \in \powerset X: x \subseteq y \implies y \in \powerset X$

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