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:
 * $\left\vert{\operatorname{im} \left({f}\right)}\right\vert = \left\vert{T}\right\vert < \left\vert{S}\right\vert$

Here, $\operatorname{Im} \left({f}\right)$ denotes the image of $f$.

Thus $\operatorname{Im} \left({f}\right) \ne S$, and so $f$ is not a bijection.