Well-Ordering Principle/Proof using Von Neumann Construction

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Every non-empty subset of $\N$ has a smallest (or first) element.

That is, the relational structure $\struct {\N, \le}$ on the set of natural numbers $\N$ under the usual ordering $\le$ forms a well-ordered set.

This is called the well-ordering principle.


From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is minimally inductive class under the successor mapping.

From Successor Mapping on Natural Numbers is Progressing, this successor mapping is a progressing mapping.

The result is a direct application of Minimally Inductive Class under Progressing Mapping is Well-Ordered under Subset Relation.