Sethood and Subclasses: Bookkeeping

This helper adopts the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, with intervals $\{m,\dots,n\}$ as in Intervals of Natural Numbers (i), the union $\cup$ as in The Boolean Operations on Classes, Disjointness, and the Universal Class (i), singletons $\{b\}$ as in The Empty Set, the Unordered Pair and the Singleton (iii), and $\subseteq$ as in Subclasses and Subsets (i).

(i) For all $m,n\in\mathbb{N}_{0}$, the interval $\{m,\dots,n\}$ is a set.

(ii) For all classes $A$ and $C$ and every set $b$, if $A\cup\{b\}\subseteq C$, then $A\subseteq C$ and $b\in C$.