Equivalence of Definitions of Well-Ordering/Definition 1 implies Definition 2

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set such that:
 * $\forall T \subseteq S: \exists a \in T: \forall x \in T: a \preceq x$

That is, such that $\preceq$ is a well-ordering by definition 1.

Then $\preceq$ is a total ordering.

Proof
Consider $X = \left\{{a, b}\right\}$ where $a, b \in S$.

By hypothesis, $X$ has a smallest element.

So either $\min X = a$ or $\min X = b$.

If $\min X = a$, then $a \preceq b$.

If $\min X = b$, then $b \preceq a$.

So either $a \preceq b$ or $b \preceq a$.

That is, $a$ and $b$ are comparable.

As this applies to all $a, b \in S$, the ordering $\preceq$ is total.