Surjection from Aleph to Ordinal

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

Suppose that:


 * $0 < y < \aleph_{x+1}$

Then there is a surjection:


 * $f : \aleph_x \to y$

Proof

 * $y < \aleph_{x+1}$, then $y < \aleph_x \lor y \sim \aleph_x$ by Ordinal Less than Successor Aleph.

In either case, $\left|{ y }\right| \le \aleph_x$ by Ordinal in Aleph iff Cardinal in Aleph and Equivalent Sets have Equal Cardinal Numbers.

The existence of the surjection follows from Surjection iff Cardinal Inequality.