Well-Ordering Principle

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

This is called the well-ordering principle.

The well-ordering principle also holds for $\N_{\ne 0}$.

Proof
The set of natural numbers is defined as the archetype of the naturally ordered semigroup.

From the definition of the naturally ordered semigroup, $\left({S, \circ, \preceq}\right)$ is well-ordered by $\preceq$.

So as $\left({\N, +, \le}\right) \cong \left({S, \circ, \preceq}\right)$ the result follows.

As $\N_{\ne 0} = \N \setminus \left\{{0}\right\}$, by Set Difference Subset $\N_{\ne 0} \subseteq \N$.

As $\N$ is well-ordered, by definition, every subset of $\N$ has a smallest element.

Also known as
This is otherwise known as the well-ordering property (of $\N$).

Some sources give it as the least-integer principle.

Note that some authors cite this as the well-ordering theorem.

However, this allows it to be confused even more easily with the Well-Ordering Theorem, which states that any set can have an ordering under which that set is a well-ordered set.

Also see

 * This theorem is equivalent to the Principle of Mathematical Induction and the Principle of Complete Induction.

Some authors extend the scope of this theorem to include:
 * Integers Bounded Below has Smallest Element
 * Integers Bounded Above has Greatest Element