Well-ordering principle for N

Well-ordering principle for $\mathbb{N}$: If $S\subseteq \mathbb{N}$ is nonempty, then there exists $m\in S$ such that $m\le n$ for all $n\in S$.

This principle is equivalent (in standard foundations) to mathematical induction and is often used to justify “choose the smallest counterexample” arguments.

Proof sketch (optional): One shows that if a nonempty set $S\subseteq\mathbb{N}$ had no least element, then by induction no natural number could belong to $S$, contradicting nonemptiness.