elementary set theory - How to prove that a set A is finite iif it is equipotent to $J_n={1,…,n}$ for some $n{in}mathbb{N}$?

Considering the (not most common) definition:

A set is infinite if it is equipotent to a proper subset of itself. A set is finite if it is not infinite.

How can I prove that a set $A$ is finite iif it is equipotent to $J_n=\{1,…,n\}$ for some $n{\in}\mathbb{N}$ (assuming that I already proved that $J_n$ is a finite set for every $n{\in}\mathbb{N}$)?

Answer

Suppose $A$ is not equipotent with any of the $J_n$. Then we see that we can produce a surjection from $A$ to $\mathbb{N}$. If not then there is some maximum $n\in \mathbb{N}$ for $f(A)$. From there you can whittle it down to a bijection to a $J_n$. Then $A$ you can conclude by contradiction that $A$ is equipotent to some $J_n$.

For the reverse direction, if $A$ is equipotent to $J_n$ for some $n$ then there is a bijection $f:A\to J_n$. Pick any $k\in J_n$ then there is no bijection from $J_n$ to $J_n\setminus \{k\}$. Since $f$ is a bijection, argue that there is no bijection from $A$ to $A\setminus\{f^{-1}(k)\}.$