Cardinality of Set is Finite iff Set is Finite

Theorem
Let $A$ be a set.


 * $\card A$ is finite


 * $A$ is finite
 * $A$ is finite

where $\card A$ denotes the cardinality of $A$.

Proof
Definition of cardinal:
 * $(1): \quad \card A \sim A$


 * $\card A$ is finite


 * $\exists n \in \N: \card A \sim \N_n$ by definition of finite set
 * $\exists n \in \N: \card A \sim \N_n$ by definition of finite set


 * $\exists n \in \N: A \sim \N_n$ by $(1)$ and Set Equivalence behaves like Equivalence Relation
 * $\exists n \in \N: A \sim \N_n$ by $(1)$ and Set Equivalence behaves like Equivalence Relation


 * $A$ is finite by definition of finite set.
 * $A$ is finite by definition of finite set.