Definition:Well-Ordered Class under Subset Relation

Definition
Let $A$ be a class which is also a nest.

Let $A$ have the property that every non-empty subclass of $A$ has a smallest element under the subset relation.

Then $A$ is said to be well-ordered under the subset relation.

Also known as
Some sources refer to the subset relation as the inclusion relation, and so the name of this property becomes well-ordered under inclusion.