The Order on Omega Is a Well-Order with Membership as Its Strict Order, and Nothing Lies between n and Its Successor

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, let $S(x)$ denote the successor of a set $x$, and let $\le$ and $<$ be the order and the strict order on $\omega$.

¶ $\le$ is a well-order on $\omega$.

¶ The class $<$ equals the strict relation associated with $\le$; in particular, for all sets $m$ and $n$, $m<n$ if and only if $m\le n$ and $m\neq n$.

¶ $0\le n$ for every $n\in\omega$.

¶ For all $m,n\in\omega$, $m<S(n)$ if and only if $m\le n$; in particular $n<S(n)$.

¶ For every $n\in\omega$ there is no $m\in\omega$ with $n<m$ and $m<S(n)$.

¶ For all $m,n\in\omega$, $m<n$ if and only if $S(m)\le n$.

¶ For every $n\in\omega$, $n=\{m\in\omega:m<n\}$.