The Set of Natural Numbers and the Number One

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 $x$.

¶ The set of natural numbers is the difference $\mathbb{N}=\omega\setminus\{0\}$; it is a set by Omega Is the Least Inductive Class: It Is a Set, Induction from Zero, the Peano Properties, and Transitivity §set and Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper §subclass. Its elements are the natural numbers.

¶ $1$ denotes the successor of $0$: $1=S(0)$.