Ordinal is Subset of Rank of Small Class iff Not in Von Neumann Hierarchy

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Let $x$ be an ordinal.

Let $S$ be a small class.

Let $\map V x$ denote the von Neumann hierarchy on the ordinal $x$.

Then $x$ is a subset of the rank of $S$ if and only if $S \notin \map V x$.



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Necessary Condition

Let $x \subseteq \map {\operatorname{rank} } S$.

Then by Von Neumann Hierarchy Comparison:

$S \in \map V x \implies S \in \map V {\map {\operatorname{rank} } S}$

But by Ordinal Equal to Rank:

$S \notin \map V {\map {\operatorname{rank} } S}$

By contraposition:

$S \notin V \left({x}\right)$


Sufficient Condition

Let $S \notin \map V x$.


\(\ds S\) \(\in\) \(\ds \map V {\map {\operatorname{rank} } S + 1}\) Ordinal Equal to Rank
\(\ds \leadsto \ \ \) \(\ds \map V {\map {\operatorname{rank} } S + 1}\) \(\nsubseteq\) \(\ds \map V x\) Rule of Transposition
\(\ds \leadsto \ \ \) \(\ds \map {\operatorname{rank} } S + 1\) \(\nsubseteq\) \(\ds x\) Von Neumann Hierarchy Comparison
\(\ds \leadsto \ \ \) \(\ds x\) \(\in\) \(\ds \map {\operatorname{rank} } S + 1\) Transitive Set is Proper Subset of Ordinal iff Element of Ordinal and Ordinal Membership is Trichotomy
\(\ds \leadsto \ \ \) \(\ds x\) \(\subseteq\) \(\ds \map {\operatorname{rank} } S\) Definition of Successor Set