Minimally Inductive Class under Progressing Mapping induces Nest/Proof 2

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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

A minimally inductive class under $g$ is the same thing as a minimally closed class under $g$ with respect to $\O$.

The result then follows by a direct application of Minimally Closed Class under Progressing Mapping induces Nest.

$\blacksquare$


Sources