Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $\omega$ and $0$ be as in The Class Omega of Natural Numbers with Zero §omega and The Class Omega of Natural Numbers with Zero §zero, $\mathbb{N}$ and $1$ as in The Set of Natural Numbers and the Number One §naturals and The Set of Natural Numbers and the Number One §one, $S(x)$ the successor of a set $x$, $+$ and $\cdot$ the addition and the multiplication on $\omega$, and $\le$ the order on $\omega$.

¶ For every set $x$, $x\in\mathbb{N}$ if and only if $x=S(m)$ for some $m\in\omega$. In particular $S(n)\in\mathbb{N}$ for every $n\in\omega$.

¶ $1\in\mathbb{N}$, and $1\le n$ for every $n\in\mathbb{N}$.

¶ $S(n)\neq1$ for every $n\in\mathbb{N}$, and for all $m,n\in\mathbb{N}$, $S(m)=S(n)$ implies $m=n$.

¶ For every $n\in\mathbb{N}$ with $n\neq1$, there is $m\in\mathbb{N}$ with $n=S(m)$.

¶ For every $n\in\omega$, $S(n)=n+1$ and $n\cdot1=n$.

¶ For all $m,n\in\mathbb{N}$, $m+n\in\mathbb{N}$ and $m\cdot n\in\mathbb{N}$.