Ordinal in Aleph iff Cardinal in Aleph

Theorem

Let $x$ and $y$ be ordinals.

Then:

$x \in \aleph_y \iff \card x \in \aleph_y$

where $\aleph$ denotes the aleph mapping.

Proof

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

By Cardinal Inequality implies Ordinal Inequality, it follows that:

$x \in \aleph_y \iff \card x \in \aleph_y$

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.46 \ (1)$