A question about proving set equality $A=B$.

Question

Suppose that $A$ and $B$ are sets such that $A\subseteq B$. Is it true that if there exist a $1-1$ function $f:B\to A$, then we may deduce that $A=B$?

Thanks!

Answer 1

No, take $A = 2\Bbb{N}, B = \Bbb{N}$ so that $A \subsetneq B$, but there exists an injective function $f:B\to A$ (defined by $f(n) = 2n$).