Definition:Strict Strong Well-Ordering

Definition
Let $A$ be a class.

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

Then $\RR$ is a strict strong well-ordering of $A$ :


 * $\RR$ connects $A$
 * $\RR$ is strongly well-founded. That is, whenever $B$ is a non-empty subclass of $A$, $B$ has a strictly minimal element under $\RR$.

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.