Lattice of Power Set is Arithmetic

Theorem
Let $X$ be a set.

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

Then $P$ is arithmetic.

Proof
Define $C = \left({K\left({P}\right), \preceq}\right)$ as an ordered subset of $P$

where $K\left({P}\right)$ denotes the compact subset of $P$.

Thus by Lattice of Power Set is Algebraic:
 * $P$ is algebraic.

It remains to prove that
 * $K\left({P}\right)$ is meet closed.

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

By definition of compact subset:
 * $x$ is compact.

By Element is Finite iff Element is Compact in Lattice of Power Set
 * $x$ is finite.

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

By Subset of Finite Set is Finite:
 * $x \cap y$ is finite.

By Element is Finite iff Element is Compact in Lattice of Power Set
 * $x \cap y$ is compact.

Thus by definition of compact subset:
 * $x \cap y \in K\left({P}\right)$