Rank is Ordinal

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Theorem

Let $S$ be a small class

The rank of $S$ is an ordinal.

Proof

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The rank of $S$ is an intersection of a set of ordinals $B$.

$B$ is nonempty by the fact that Every Set in Von Neumann Universe.

Thus, $B$ has a minimal element, which is the rank of $S$ plus $1$.

Therefore, the rank is itself an ordinal.

$\blacksquare$