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.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Definition $4.8$