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
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 2$ Superinduction and double superinduction: Theorem $2.4 \ (2)$