Every Set in Von Neumann Universe

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Let $S$ be a small class.

Then $S$ is well-founded.



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The proof shall proceed by Epsilon Induction on $S$.

Suppose that all the elements $a \in S$ are well-founded.

That is, $a \in V \left({x}\right)$ for some $x$.


$\displaystyle F \left({a}\right) = \bigcap \left\{{x \in \operatorname{On} : a \in V \left({x}\right)}\right\}$

Take $\displaystyle \bigcup_{a \mathop \in S} F \left({a}\right)$.

Take any $a \in S$.

\(\displaystyle a\) \(\in\) \(\displaystyle V \left({F \left({a}\right)}\right)\) $\quad$ Definition of $F$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle a\) \(\in\) \(\displaystyle V \left({\bigcup_{x \mathop \in S} F \left({x}\right) }\right)\) $\quad$ Set is Subset of Union: Family of Sets and Von Neumann Hierarchy Comparison $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle S\) \(\subseteq\) \(\displaystyle V \left({\bigcup_{x \mathop \in S} F \left({x}\right) }\right)\) $\quad$ Definition of Subset $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle S\) \(\in\) \(\displaystyle \mathcal P \left({ V\left({ \bigcup_{x \mathop \in S} F\left({x}\right) }\right) }\right)\) $\quad$ Definition of Power Set $\quad$


$\displaystyle S \in V \left({ \bigcup_{x \mathop \in S} F \left({x}\right) + 1}\right)$

and $S \in V \left({x}\right)$ for some ordinal $x$.