Well-ordering principle for the natural numbers ($S\subseteq \mathbb{N},S\ne \varnothing \Rightarrow \min S\text{ exists}$)

Every nonempty subset of the natural numbers has a least element in their usual order. Applying this principle to positive integer linear combinations proves Bézout's identity. This principle for the usual natural-number order is distinct from the general well-ordering theorem asserting that every set can be equipped with some well-order.