Principal Ultrafilter is All Sets Containing Cluster Point

Theorem
Let $S$ be a set.

Let $\powerset S$ denote the power set of $S$.

Let $\FF \subset \powerset S$ be a principal ultrafilter on $S$.

Let its cluster point be $x$.

Then $\FF$ is the set of all subsets $T$ of $S$ such that $x \in T$.

Proof
We have that the cluster point of $\FF$ is $x$.

there is some $A \subseteq S$ such that $x \in A$ and $A \notin \FF$.

Then, by definition of relative complement:
 * $x \notin \relcomp S A$

By definition of cluster point, $x$ is in every element of $\FF$.

But as $x \notin \relcomp S A$, it follows that:
 * $\relcomp S A \notin \FF$

This contradicts $\FF$ being an ultrafilter:
 * for every $A \subseteq S$, either $A \in \FF$ or $\relcomp S A \in \FF$

Therefore, by Proof by Contradiction, every $A \subseteq S$ such that $x \in A$ is in $\FF$.

Thus, $A \subseteq S$ is in $\FF$ precisely when $x \in A$.