The Number n Lies in the Segment [n], and Membership in the Addition Class on ω

This helper adopts The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, with $[n]$ as in Intervals of Natural Numbers (ii). Let $\omega=\mathbb{N}_{0}$ and let $+$ be the class of Addition on Omega (i).

(i) For every $n\in\mathbb{N}$, $n\in[n]$.

(ii) For all sets $a$, $b$ and $c$, $((a,b),c)\in{+}$ if and only if $a,b\in\omega$ and $c=a+b$.