Principal Ultrafilter is All Sets Containing Cluster Point

Theorem
Let $X$ be a set, and $\mathcal P \left({X}\right)$ be the power set of $X$.

Let $\mathcal F \subset \mathcal P \left({X}\right)$ be a principal ultrafilter on $X$.

Let its cluster point be $x$.

Then $\mathcal F$ is the set of all subsets of $X$ which contain $x$.