No Bijection between Finite Set and Proper Subset/Proof 1

Theorem

A finite set can not be in one-to-one correspondence with one of its proper subsets.

That is, a finite set is not Dedekind-infinite.

Proof

Let $S$ be a finite set.

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:

$\card {\Img f} = \card T < \card S$

where $\Img f$ denotes the image of $f$.

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

$\blacksquare$