Set is Countable iff Cardinality not greater Aleph Zero

Theorem

Let $X$ be a set.

Then:

$X$ is countable if and only if $\size X \le \aleph_0$

where

$\size X$ denotes the cardinality of $X$,

$\aleph$ denotes the aleph mapping.

Proof

$X$ is countable

if and only if

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

if and only if

$\size X \le \size \N$ by Injection iff Cardinal Inequality

if and only if

$\size X \le \aleph_0$ by Aleph Zero equals Cardinality of Naturals.

$\blacksquare$