Definition:Well-Ordering/Class Theory

Definition
Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a total ordering.

Then $\RR$ is a well-ordering :
 * every non-empty subclass of $\Field \RR$ has a smallest element under $\RR$

where $\Field \RR$ denotes the field of $\RR$.

Also known as
There is a school of thought that suggests that this may be referred to as a strong well-ordering, keeping the terminology different from that of a well-ordering.

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.