The Principle of Induction for the Natural Numbers Starting at One

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $\mathbb{N}$ and $1$ be as in The Set of Natural Numbers and the Number One §naturals and The Set of Natural Numbers and the Number One §one, and let $S(x)$ denote the successor of a set $x$.

¶ Let $A$ be a class with $A\subseteq\mathbb{N}$ such that $1\in A$ and $S(n)\in A$ for every $n\in A$. Then $A=\mathbb{N}$.