# Aleph is Infinite Cardinal

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## Theorem

Let $x$ be an ordinal.

Then $\aleph_x$ is an infinite cardinal where $\aleph$ denotes the aleph mapping.

## Proof

By the definition of the aleph mapping:

- $\aleph : \operatorname{On} \to \mathcal N'$

The theorem statement is an immediate consequence of this fact.

$\blacksquare$