Countable iff Cardinality not greater than Aleph Zero

Theorem
Let $X$ be set.

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

where:
 * $\left\vert{X}\right\vert$ denotes the cardinality of $X$
 * $\aleph_0 = \left\vert{\N}\right\vert$ by Aleph Zero equals Cardinality of Naturals.

Proof

 * $X$ is countable


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


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


 * $\left\vert{X}\right\vert \leq \aleph_0$
 * $\left\vert{X}\right\vert \leq \aleph_0$