Sandwich Principle for G-Towers/Corollary 1

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Theorem

Let $M$ be a class.

Let $g: M \to M$ be a progressing mapping on $M$.

Let $M$ be a $g$-tower.


Let $x, y \in M$ be arbitrary.

Then:

$x \subsetneqq y \implies \map g x \subseteq y$


Proof

From Lemma $2$ of $g$-Tower is Nest we have that:

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


From the Sandwich Principle: Corollary 1:

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


That is, if:

$x \subsetneqq y$

then:

$\map g x \subseteq y$

$\blacksquare$


Also see


Sources