TITLE: X is infinite if and only if X is equivalent to a proper subset of itself.
QUESTION [7 upvotes]: Prove that a set $X$ is infinite if and only if $X$ is equivalent to a proper subset of itself.

If $X$ is finite, then suppose $|X|=n$. Any proper subset $Y$ of $X$ has size $m<n$, and so there cannot be any bijective mapping between $Y$ and $X$.
If $X$ is countably infinite, then suppose $X=\{x_1,x_2,\ldots\}$. We can map $X$ to $Y=\{x_2,x_3,\ldots\}$ by using the map $f(x_i)=x_{i+1}$.
But what if $X$ is uncountably infinite? How can we specify the mapping?

REPLY [2 votes]: The answer is no if you do not assume the axiom of choice or a weaker version such as countable choice.  If $X$ is an amorphous set, then it is Dedekind-finite, which is the negation of the property in the question.