Ordinal in Aleph iff Cardinal in Aleph

Theorem
Let $x$ and $y$ be ordinals.

Then:


 * $x \in \aleph_y \iff \left|{ x }\right| \in \aleph_y$ where $\aleph$ denotes the aleph mapping.

Proof
By the definition of the aleph mapping, $\aleph_y$ is an element of the infinite cardinal class.

By Cardinal Inequality implies Ordinal Inequality, it follows that:


 * $x \in \aleph_y \iff \left|{ x }\right| \in \aleph_y$