Definition:Strongly Well-Founded Relation

Definition
Let $A$ be a class.

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

Then $A$ is well-founded iff whenever $B$ is a non-empty subclass of $A$, $B$ has an $\mathcal R$-minimal element.

Also known as
A well-founded relation may also be called foundational, but that is used in a weaker sense on.

According to Well-Founded Relation Determines Minimal Elements, the Axiom of Foundation implies that every foundational relation is well-founded.

Remarks
As the definition quantifies over general subclasses, well-foundedness is not a first-order property. Thus this property cannot be used with the axiom of specification to specify classes in von Neumann-Bernays-Gödel (NBG) class-set theory, although it can be used so in Morse-Kelley class-set theory.