Multiplication on Omega: Products Lie in Omega, Multiplication Is a Binary Operation with Value the Product, and Membership in the Multiplication

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation, let $\omega$ be as in The Class Omega of Natural Numbers with Zero §omega, and let $\cdot$ be the multiplication on $\omega$, with the product $m\cdot n$ of $m,n\in\omega$ as defined there.

¶ For all $m,n\in\omega$, $m\cdot n\in\omega$.

¶ The class $\cdot$ is a binary operation on $\omega$, and for all $m,n\in\omega$ its value at $(m,n)$ equals the product $m\cdot n$; so the notation of Binary Operations on a Set §notation for $\cdot$ agrees with the product.

¶ For all sets $m$, $n$ and $z$: $((m,n),z)\in{\cdot}$ if and only if $m\in\omega$, $n\in\omega$ and $z=m\cdot n$.