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 non-empty by the fact that Set has Rank.
Thus, $B$ has a minimal element, which is the rank of $S$ plus $1$.
Therefore, the rank is itself an ordinal.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 9.15 \ (1)$