I want to prove Schröder–Bernstein theorem for 2 finite sets $A, B$ of the same cardinal. I do it with induction on the cardinal number of $A, B$. In the inducative step, I write:
Let $A, B$ sets such that $|A|=|B|=n+1$ for some $n\in\omega$. Here, I want to use the the inducative assumption for the sets $A-\{a\}, B-\{b\}$ for some $a\in A, b\in B$. For choosing the above $a,b$, do I need to use the axiom of choice?