Aleph is Infinite Cardinal

Theorem
Let $x$ be an ordinal.

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

Proof
Let $\On$ be the ordinal class.

By definition of the aleph mapping:
 * $\aleph: \On \to \NN'$

where $\NN'$ is the class of infinite cardinal numbers.

The theorem statement is an immediate consequence of this fact.