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.

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

By Empty is Bottom of Lattice of Power Set:
 * $\varnothing = \bot$

where $\bot$ denotes the bottom of $L$.

By Bottom is Way Below Any Element:
 * $\bot \ll \bot$

where $\ll$ is the way below relation.

By definition:
 * $\bot$ is compact.

By definition of the smallest element:
 * $\bot \preceq x$

By definition of compact closure:
 * $\bot \in x^{\mathrm{compact} }$

By definition:
 * $x^{\mathrm{compact} }$ is non-empty.

Thus by Non-Empty Compact Closure is Directed:
 * $x^{\mathrm{compact} }$ 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 $x \in \mathcal P\left({X}\right)$.

We will prove that
 * $\forall a \in \mathcal P\left({X}\right): a$ is upper bound for $x^{\mathrm{compact} } \implies x \preceq a$

Let $a \in \mathcal P\left({X}\right)$ such that
 * $a$ is upper bound for $x^{\mathrm{compact} }$

We will prove that
 * $x \subseteq a$

Let $t \in x$.

By definition of power set:
 * $x \subseteq X$

By definitions of subset and singleton:
 * $\left\{ {t}\right\} \subseteq X$ and $\left\{ {t}\right\} \subseteq x$

By definition of power set:
 * $\left\{ {t}\right\} \in \mathcal P\left({X}\right)$

By Singleton is Finite:
 * $\left\{ {t}\right\}$ is finite.

By definition of $\mathit{Fin}$:
 * $\left\{ {t}\right\} \in \mathit{Fin}\left({x}\right)$

where $\mathit{Fin}\left({x}\right)$ denotes the set of all finite subset of $x$.

By Compact Closure is Set of Finite Subsets in Lattice of Power Set:
 * $\left\{ {t}\right\} \in x^{\mathrm{compact} }$

By definition of upper bound:
 * $\left\{ {t}\right\} \preceq a$

By definition of $\preceq$:
 * $\left\{ {t}\right\} \subseteq a$

Thus by definitions of subset and singleton:
 * $t \in a$

Thus by definition of $\preceq$:
 * $x \preceq a$

By definition of compact closure:
 * $\forall y \in x^{\mathrm{compact} }: y \preceq x$

By definition of upper bound:
 * $x$ is upper bound for $x^{\mathrm{compact} }$

Thus by definition of supremum:
 * $x = \sup \left({x^{\mathrm{compact} } }\right)$