Lattice of Power Set is Algebraic

Theorem
Let $X$ be a set.

Let $L = \left({\mathcal P\left({X}\right), \cup, \cap, \preceq}\right)$ be the lattice of power set of $X$ where $\mathord\preceq = \mathord\subseteq \cap \left({\mathcal P\left({X}\right) \times \mathcal P\left({X}\right)}\right)$

Then $L$ is algebraic.

Proof
We will prove that
 * $\forall x \in \mathcal P\left({X}\right): x^{\mathrm{compact} }$ is directed.

By Power Set is Complete Lattice:
 * $L$ is complete lattice.

Thus by definition of complete lattice:
 * $L$ is up-complete.

It remains to prove that
 * $L$ satisfies axiom of K-approximation.

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