Well-ordering principle

Statement

Every nonempty set of natural numbers has a least element. That is, if $A \subseteq \mathbb{N}$ and $A \neq \emptyset$, then there is an $a$ in $A$ with $a \le x$ for each $x$ in $A$.

Remarks

The least element is unique, because the order on $\mathbb{N}$ is antisymmetric. The principle follows from strong induction, and conversely it implies the principle of mathematical induction. The set of the integers does not have this property: the set of negative integers has no least element.

Explanations

L1 human

If some natural numbers have a property, then there is a smallest one with that property. For example, among the natural numbers that are larger than $10$ and even, the smallest is $12$. Proofs often use this: they take the smallest counterexample and show that it cannot exist.