Question on one to one correspondence implies same cardinality, for finite sets I have been taking for granted the fact that given two finite sets $A$ and $B$, if there exists a bijective map then the number of elements in $A$ and $B$ are the same, because I thought this was a trivial fact.  
But then the same doesn't really hold (in some sense) when $A$ and $B$ have infinite elements, because if $A = 2 \mathbb{N}$ and $B = \mathbb{N}$ surely there is a bijective map from $A$ to $B$, but $A$ is a strict subset of $B$. 
This made me think that perhaps in the proof of the fact that given two finite sets $A$ and $B$ the existence of a bijective map implies $|A| = |B|$, the assumption that $|A|$ and $|B|$ are finite must be used.
I was wondering if someone could tell me how to prove this fact(this very basic fact!) and point to me where the assumption is being used. Thank you very much!

Remember, the statement

$|A|=|B|$

is just shorthand for the statement

"There exists a bijection $f\colon A \to B$."

In particular, as there is a bijection $f\colon 2\mathbb{N} \to \mathbb{N}$, it follows from definition that $|2\mathbb{N}|=|\mathbb{N}|$.
Infinite sets of the same cardinality can be properly contained within one another. What you may have noticed, however, is if $A\subseteq B$ are finite sets, and $|A|=|B|$, then $A = B$.
