Definition:Well-Founded Relation/Warning

Well-Founded Relation: Warning
According to Strictly Well-Founded Relation is Well-Founded, the Axiom of Foundation implies that every strictly well-founded relation is well-founded.

As the definition quantifies over general subclasses, well-foundedness is not a first-order property.

Thus it cannot be used in a formula of a theory, like NBG class-set theory, that only allow quantification over sets.

For example it cannot be used with the axiom schema of specification in NBG set theory, and statements using it must generally be framed as schemas.

A higher-order theory such as Morse-Kelley set theory allows it to be used more freely and directly.

A concrete example:

Let $A$ be a class.

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

One cannot directly use the axiom schema of specification in NBG theory to form the class of all subsets $x$ of $A$ such that the restriction of $\RR$ to $A \setminus x$ is a well-founded relation.