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 $\On$ denote the class of all ordinals.

By definition of the aleph mapping:

$\aleph: \On \to \NN'$

where $\NN'$ denotes the class of infinite cardinals.

The theorem statement is an immediate consequence of this fact.

$\blacksquare$