Minimally Inductive Class under Progressing Mapping induces Nest

Theorem
Let $M$ be a class which is minimally inductive under a progressing mapping $g$.

Then $M$ is a nest in which:
 * $\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$

Proof
Let $\RR$ be the relation on $M$ defined as:


 * $\forall x, y \in M: \map \RR {x, y} \iff \map g x \subseteq y \lor y \subseteq x$

The fact that:
 * $\map g x \subseteq y \lor y \subseteq x$

implies that:
 * $x \subseteq y \lor y \subseteq x$

because:
 * $x \subseteq \map g x$

The result follows as a direct application of the Progressing Function Lemma.