Class of Finite Character is Swelled

Theorem

Let $A$ be a class which has finite character.

Then $A$ is a swelled class.


Proof

Let $x \in A$ and $y \subseteq x$.

Then by hypothesis every finite subset of $y$ is also a finite subset of $x$.

Hence every finite subset of $y$ is in $A$.

Hence again by hypothesis $y \in A$.

$\blacksquare$


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: $\S 5$ Maximal principles: Lemma $5.4 \ (1)$
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