Ordering where every Subclass has Smallest Element is Well-Ordering

Theorem
Let $A$ be a class.

Let $\RR$ be an ordering on $A$.

Let $\RR$ be such that every subclass of $A$ has a smallest element under $\RR$.

Then $\RR$ is a well-ordering.

Proof
We are given that $\RR$ is an ordering.

It will be shown that $\RR$ is a total ordering, from which the fact that it is a well-ordering follows directly.

Let $x$ and $y$ be elements of $A$.

Then $\set {x, y}$ is a subclass of $A$.

Hence $\set {x, y}$ has a smallest element.

That is, either $x \mathrel \RR y$ or $y \mathrel \RR x$.

As $x$ and $y$ are arbitrary, it follows that:
 * $\forall x, y \in A: x \mathrel \RR y \lor y \mathrel \RR x$

Hence, by definition, $\RR$ is a total ordering.

So, we have that $\RR$ is a total ordering on $A$ such that every subclass of $A$ has a smallest element under $\RR$.

That is, $\RR$ is a well-ordering.