Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle

Theorem

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

Then for all $x, y \in M$:

$x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$

Proof 1

From Minimally Inductive Class under Progressing Mapping induces Nest, we have that $M$ is a nest in which:

$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$

Thus the Sandwich Principle applies directly.

$\blacksquare$

Proof 2

By definition of minimally inductive class, $M$ is minimally closed under $g$ with respect to $\O$.

The result is then seen to be a direct application of Sandwich Principle for Minimally Closed Class.

$\blacksquare$