Element is Finite iff Element is Compact 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 lattice of power set.

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

Then $x$ is a finite set $x$ is a compact element.