Filter is Finer iff Sets of Basis are Subsets

Theorem
Let $S$ be a set.

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

Let $\FF, \FF' \subset \powerset S$ be two filters on $S$.

Let $\FF$ have a basis $\BB$.

Let $\FF'$ have a basis $\BB'$.

$\FF$ is finer than $\FF'$ every set of $\BB$ is a subset of a set of $\BB'$.