Class under Progressing Mapping such that Elements are Sandwiched is Nest
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Theorem
Let $A$ be a class.
Let $g: A \to A$ be a progressing mapping on $A$ such that:
- $\forall x, y \in A: \map g x \subseteq y \lor y \subseteq x$
Then $A$ is a nest:
- $\forall x, y \in A: x \subseteq y \lor y \subseteq x$
Proof
By definition of progressing mapping:
- $\forall x \in A: x \subseteq \map g x$
Thus by Subset Relation is Transitive:
- $\map g x \subseteq y \implies x \subseteq y$
and it follows that:
- $\forall x, y \in A: x \subseteq y \lor y \subseteq x$
Hence the result by definition of nest.
$\blacksquare$
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: Lemma $4.9$