Looking for Cantor's original proof of the Cantor-Bernstein theorem that relies on the axiom of choice? Even a sketch of it would be good enough. Thanks.

Using the axiom of choice, every set can be well ordered. For each set $X$, let $|X|$ be the smallest ordinal order-isomorphic to some well-ordering of $X$. If there is an injection $f:Y\to X$,  then $Y$ can be identified with a subset of $X$ and hence with a subset of $|X|$. The order type of this set is smaller or equal to $|X|$, hence $|Y|\leq |X|$. If there is also an injection from $X$ to $Y$, we can conclude that $|X|\leq|Y|$. Since the class of ordinals is well-ordered it satisfies anti-symmetry and we can conclude that $|X|=|Y|$. So there are bijections from both $X$ and $Y$ to the ordinal $|X|=|Y|$, and combining these bijections gives us a bijection between $X$ and $Y$.  
