Set is Countable iff Cardinality not greater Aleph Zero

Theorem
Let $X$ be a set.

Then:
 * $X$ is countable $\left\vert{X}\right\vert \leq \aleph_0$

where
 * $\left\vert{X}\right\vert$ denotes the cardinality of $X$,
 * $\aleph$ denotes the aleph mapping.

Proof

 * $X$ is countable


 * there exists an injection $X \to \N$ by definition of countable set


 * $\left\vert{X}\right\vert \leq \left\vert{\N}\right\vert$ by Injection iff Cardinal Inequality


 * $\left\vert{X}\right\vert \leq \aleph_0$ by Aleph Zero equals Cardinality of Naturals.