Element is Finite iff Element is Compact in Lattice of Power Set

Theorem
Let $X$ be a set.

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

Let $x \in \powerset X$.

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

Sufficient Condition when Empty
The case when $x = \O$

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

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

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

where $\ll$ denotes the way below relation.

Thus by definition
 * $x$ is a finite set $x$ is a compact element.

Sufficient Condition when Non-empty
The case when $x \ne \O$:

Let $x$ be a finite set.

We will prove that
 * for every a set $Y$ of subsets of $X$ such that $x \subseteq \bigcup Y$
 * then there exists a finite subset $Z$ of $Y$: $x \subseteq \bigcup Z$

Let $Y$ be a set of subsets of $X$ such that
 * $x \subseteq \bigcup Y$

By definitions of union and subset:
 * $\forall e \in x: \exists u \in Y: e \in u$

By Axiom of Choice:
 * $\exists f:x \to Y: \forall e \in x: e \in \map f e$

Define:
 * $Z = f \sqbrk x$

By Image of Mapping from Finite Set is Finite:
 * $Z$ is a finite set.

By definition of image of mapping:
 * $Z \subseteq Y$

It remains to prove that
 * $x \subseteq \bigcup Z$

Let $e \in x$.

By definitions of $f$ and image of mapping:
 * $e \in \map f e \in Z$

Thus by definition of union:
 * $e \in \bigcup Z$

By Way Below in Lattice of Power Set:
 * $x \ll x$

Thus by definition of compact:
 * $x$ is compact.

Necessary Condition when Non-empty
Let $x$ be a compact element.

By definition of subset:
 * $Y := \set {\set y: y \in x}$ is set of subsets of $X$.

We will prove that
 * $x \subseteq \bigcup Y$

Let $y \in x$.

By definition of $Y$:
 * $\set y \in Y$

By definition of singleton:
 * $y \in \set y$

Thus by definition of union:
 * $y \in \bigcup Y$

By definition of compact:
 * $x \ll x$

By Way Below in Lattice of Power Set:
 * there exists finite subset $Z$ of $Y$: $x \subseteq \bigcup Z$

By Singleton is Finite:
 * $\forall A \in Z: A$ is a finite set.

By Finite Union of Finite Sets is Finite:
 * $\bigcup Z$ is a finite set.

Thus by Subset of Finite Set is Finite:
 * $x$ is a finite set.