Filter is Finer iff Sets of Basis are Subsets

Theorem
Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ denote the power set of $S$.

Let $\mathcal F, \mathcal F' \subset \mathcal P \left({S}\right)$ be two filters on $S$.

Let $\mathcal F$ have a basis $\mathcal B$.

Let $\mathcal F'$ have a basis $\mathcal B'$.

$\mathcal F$ is finer than $\mathcal F'$ every set of $\mathcal B$ is a subset of a set of $\mathcal B'$.