# Definition:Well-Ordered Class under Subset Relation

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## 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 $4.8$