No Bijection between Finite Set and Proper Subset/Proof 1

Proof
Let $S$ be a finite set, and let $T$ be a proper subset of $S$.

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

By Cardinality of Image of Injection and Cardinality of Subset of Finite Set:
 * $\size {\Img f} = \size T < \size S$

Here, $\Img f$ denotes the image of $f$.

Thus $\Img f \ne S$, and so $f$ is not a bijection.