Omega Is the Least Inductive Class: It Is a Set, Induction from Zero, the Peano Properties, and Transitivity

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, and let $S(x)$ denote the successor of a set $x$.

¶ $\omega$ is a set.

¶ $\omega$ is inductive: $0\in\omega$, and $S(n)\in\omega$ for every $n\in\omega$.

¶ Every inductive class $A$ satisfies $\omega\subseteq A$.

¶ Let $A$ be a class such that $0\in A$ and, for every $n\in\omega$, $n\in A$ implies $S(n)\in A$. Then $\omega\subseteq A$.

¶ $S(x)\neq0$ for every set $x$.

¶ For all sets $x$ and $y$, $S(x)=S(y)$ implies $x=y$.

¶ For every $n\in\omega$, either $n=0$ or there is $m\in\omega$ with $n=S(m)$.

¶ For every $n\in\omega$ and every set $u$, if $u\in n$ then $u\in\omega$.

¶ For every $n\in\omega$ and all sets $u$ and $v$, if $v\in u$ and $u\in n$ then $v\in n$.