Injection iff Cardinal Inequality

Theorem
Let $\left|{ T }\right|$ denote the cardinal number of $T$.

Let $S$ and $T$ be sets such that $S \sim \left|{ S }\right|$ and $T \sim \left|{ T }\right|$.

Then:


 * $\left|{ S }\right| \le \left|{ T }\right|$ iff there is an injective mapping $f : S \to T$.

Necessary Condition
Suppose that $\left|{ S }\right| \le \left|{ T }\right|$.

Let $g : S \to \left|{ S }\right|$ be a bijection and $h : \left|{ T }\right| \to T$ be a bijection.

It follows that $g : S \to \left|{ T }\right|$ is an injection by the fact that $\left|{ T }\right| \le \left|{ S }\right|$.

Then, $h \circ g : S \to T$ is an injection.

Sufficient Condition
The other direction follows from Injection implies Cardinal Inequality.