Element of Swelled Set with no Immediate Extension is Maximal
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Theorem
Let $S$ be a swelled set of sets.
Let $x \in S$ have no immediate extension.
Then $x$ is a maximal element of $S$ with respect to the subset relation.
Proof
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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 $\text {II}$ -- Maximal principles: $\S 6$ Another approach to maximal principles