Non-Empty Class has Element of Least Rank

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Theorem

Let $C$ be a class.

Let $C \ne \varnothing$.

Then $C$ has an element of least rank.

That is:

$\exists x \in C: \forall y \in C: \operatorname{rank} \left({x}\right) \le \operatorname {rank}\left({y}\right)$

where $\operatorname{rank}\left({x}\right)$ is the rank of $x$.


Proof

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By Set has Rank, each element of $C$ has a rank.

Let $R$ be the class of ranks of elements of $C$.

$R$ is non-empty because $C$ is non-empty.



Since any non-empty class of ordinals has a least element, $R$ has a least element, $q$.

By the definition of $R$:

$\exists x \in C: \operatorname{rank}\left({x}\right) = q$

Then $x$ is an element of $C$ of least rank.

$\blacksquare$