Well-Ordering on Set is Proper Well-Ordering

Theorem
Let $\struct {S, \preccurlyeq}$ be a well-ordered set.

Then $\preccurlyeq$ is a proper well-ordering.

Proof
By definition, a proper well-ordering is a well-ordering on a class such that:


 * every proper lower section of $S$ is a set.

We have that a proper lower section of $S$ is a subclass of $S$.

But here we have that $S$ is a set.

The result follows from Subclass of Set is Set.