Definition:Strict Strong Well-Ordering

Definition
Let $A$ be a class.

Let $\mathcal R$ be a relation on $A$.

Then $\mathcal R$ is a strict strong well-ordering of $A$ iff:


 * $\mathcal R$ connects $A$;
 * $\mathcal R$ is strongly well-founded. That is, whenever $B$ is a non-empty subclass of $A$, $B$ has an $\mathcal R$-minimal element.

Also known as
calls this a well-ordering, but we use that term in a slightly different sense.

Remarks
The term "strict strong well-ordering" was invented for to distinguish between this notion and the weaker notion of a strict well-ordering. In the presence of the Axiom of Foundation, they are equivalent.