I am reading Hrbacek-Jech's Introduction to Set Theory (3ed), and in chapter 4, section 2, which deals with finite sets, they define the following:
(Remember: in set theory, $n$ is a natural number iff $n = \{0, 1, \dots, n-1 \}$)
Definition: A set $X$ is finite if there is a bijection between $X$ and a natural number $n$; in such case we define the cardinal number of $X$ to be $\lvert X \rvert = n$. Moreover, a set is said to be infinite if it is not finite.
Then comes the first result of the section:
Theorem: If $n$ is a natural number, then there is no bijection between $n$ and a proper subset $X \subset n$.
Immediately after they prove $\mathbb{N}$ is an infinte set using the stronger result:
Theorem: If $X$ is a finite set, then there is no bijection between $X$ and a proper subset $Y \subset X$.
My problem is that this stronger result is never proved, and is only alluded to at the very end of the section (literally in the very last paragraph), as a mere consequence of the weaker result above. However I've tried to prove it using only what is given up to that point but have failed. Considering the cursory treatment they gave to it I imagine the proof is a one-liner, but so far I have come empty handed.
So my question is: how do you prove this stronger result?
Best Answer
Take a bijection $\varphi$ between $X$ and a natural number $n$.
Now suppose that $\psi$ is a bijection between $X$ and a proper subset $Y$.
$\varphi \circ \psi \circ \varphi^{-1}$ is a bijection between $n$ and a proper subset. A contradiction.