Countable Choice Follows from Dependent Choice

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, assume the axiom of dependent choice, and let $\mathbb{N}$ be as in The Set of Natural Numbers and the Number One §naturals.

¶ Let $b$ be a set and let $(A_{m})_{m\in\mathbb{N}}$ be a family indexed by $\mathbb{N}$ such that, for every $m\in\mathbb{N}$, $A_{m}\subseteq b$ and $A_{m}$ has at least one element. Then there is a map $x:\mathbb{N}\to b$, written $(x_{m})_{m\in\mathbb{N}}$, such that $x_{m}\in A_{m}$ for every $m\in\mathbb{N}$.