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, $a \in \map V x$ for some $x$.


$\ds \map F a = \bigcap \set {x \in \On : a \in \map V x}$

Take $\ds \bigcup_{a \mathop \in S} \map F a$.

Take any $a \in S$.

\(\ds a\) \(\in\) \(\ds \map V {\map F a}\) Definition of $F$
\(\ds \leadsto \ \ \) \(\ds a\) \(\in\) \(\ds \map V {\bigcup_{x \mathop \in S} \map F x}\) Set is Subset of Union: Family of Sets and Von Neumann Hierarchy Comparison
\(\ds \leadsto \ \ \) \(\ds S\) \(\subseteq\) \(\ds \map V {\bigcup_{x \mathop \in S} \map F x}\) Definition of Subset
\(\ds \leadsto \ \ \) \(\ds S\) \(\in\) \(\ds \powerset {\map V {\bigcup_{x \mathop \in S} \map F x} }\) Definition of Power Set


$\ds S \in \map V {\bigcup_{x \mathop \in S} \map F x + 1}$

and $S \in \map V x$ for some ordinal $x$.