Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle/Proof 1
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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
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.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Theorem $4.10 \ (1)$