Set has Rank

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Theorem

Let $S$ be a set.


Then $S$ has a rank.


Proof

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Let $G$ be the smallest transitive set containing $S$ as a subset.

By Set Contained in Smallest Transitive Set, $G$ must exist.

By Transitive Set Contained in Von Neumann Hierarchy Level, $G \subseteq V_i$ for some ordinal $i$.

Therefore:

$G \in V_{i+1}$



We have that the ordinals are well-ordered.

From Proper Well-Ordering determines Smallest Elements, there exists a smallest ordinal $k$ such that $G \in V_{k+1}$.

Hence, by definition, $G$ has rank $k$.

$\blacksquare$