Characteristics of Minimally Inductive Class under Progressing Mapping/Image of Proper Subset is Subset
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Theorem
Let $M$ be a class which is minimally inductive under a progressing mapping $g$.
Then for all $x, y \in M$:
- $x \subset y \implies \map g x \subseteq y$
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 corollary $1$ of 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 $4.10 \ (2)$