Cardinality of Set is Finite iff Set is Finite

Theorem
Let $A$ be a set.

$\left\vert{A}\right\vert$ is finite

$A$ is finite.

Proof
Definition of cardinal:
 * $(1): \quad \left\vert A \right\vert \sim A$.

$\left\vert{A}\right\vert$ is finite

$\iff$ $\exists n \in \N: \left\vert A \right\vert \sim \N_n$ by definition of finite set

$\iff$ $\exists n \in \N: A \sim \N_n$ by $(1)$ and Set Equivalence is Equivalence Relation

$\iff$ $A$ is finite by definition of finite set.