definition of well ordered set, a variant

A well-ordered set is normally defined as a totally ordered set in which every nonempty subset has a least member, as the parent object does.

It is possible to define well-ordered sets as follows:

a well-ordered set $X$ is a partially ordered set in which every nonempty subset of $X$ has a least member.

To justify the alternative, we prove that every partially ordered set $X$ in which every nonempty subset has a least member is total:

let $x\in X$ and $y\in X$, $x\ne y$. Now, $\{x,y\}$ has a least member, a fortiori, $x,y$ are comparable. Hence, $X$ is totally ordered.

The alternative has the benefit of being a stronger statement in the sense that

$$(partialorder)⟹(totalorder)$$

given that every nonempty subset has a least member.