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

## Theorem

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 $S \notin \map V x$.

## Proof

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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:

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

$\Box$

### Sufficient Condition

Let $S \notin \map V x$.

Then:

 $\ds S$ $\in$ $\ds \map V {\map {\operatorname{rank} } S + 1}$ 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

$\blacksquare$