Sandwich Principle/Corollary 1
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Theorem
Let $A$ be a class.
Let $g: A \to A$ be a mapping on $A$ such that:
- for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$.
Let:
- $x \subset y$
where $\subset$ denotes a proper subset.
Then:
- $\map g x \subseteq y$
Proof
Let $x \subset y$.
By hypothesis, either $\map g x \subseteq y$ or $y \subseteq x$.
But because $x \subset y$, it follows that $y \subseteq x$ cannot be the case.
Hence the result.
$\blacksquare$
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 $4.9 \ (2)$