Definition:Strong Well-Ordering

Definition
Let $A$ be a class.

Let $\preceq$ be a relation on $A$.

Then $\preceq$ is a strong well-ordering :


 * $\preceq$ is an ordering.


 * For each non-empty subclass $B \subseteq A$, $B$ has a smallest element with respect to $\preceq$.

Remarks
call this a well-ordering, but we use that term in a weaker sense. Thus to avoid ambiguity, we have invented the term strong well-ordering for use on.

The difference is that for an ordering to be a well-ordering, every non-empty subset must have a smallest element, whereas for an ordering to be a strong well-ordering, not only every subset but in fact every subclass must have a smallest element.