Injection implies Cardinal Inequality

Theorem
Let $S$ and $T$ be sets.

Let $f: S \to T$ be an injection.

Let $\card T$ denote the cardinal number of $T$.

Let:
 * $T \sim \card T$

where $\sim$ denotes set equivalence

Then:
 * $\card S \le \card T$

Proof
Let $f \sqbrk S$ denote the image of $S$ under $f$.