Membership Rank Inequality
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Theorem
Let $S$ and $T$ be sets.
Let $\map {\operatorname{rank} } S$ denote the rank of $S$.
Then:
- $S \in T \implies \map {\operatorname{rank} } S < \map {\operatorname{rank} } T$
Proof
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- $T \in \map V {\map {\operatorname{rank} } T + 1}$
By the definition of rank:
- $T \subseteq \map V {\map {\operatorname{rank} } T}$
Since $S \in T$:
- $S \in \map V {\map {\operatorname{rank} } T}$
By Ordinal is Subset of Rank of Small Class iff Not in Von Neumann Hierarchy:
- $\map {\operatorname{rank} } T \nsubseteq \map {\operatorname{rank} } S$
Therefore by Ordinal Membership is Trichotomy and Transitive Set is Proper Subset of Ordinal iff Element of Ordinal:
- $\map {\operatorname{rank} } S < \map {\operatorname{rank} } T$
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 9.16$