This assertion is not provable in ZFC (if ZFC is consistent, that is). If ZFC is consistent then it has a model in which this assertion is true, for example in Godel's constructible universe. However by forcing we can make the following true:
Which is a clear contradiction to the statement that $\kappa\mapsto 2^\kappa$ is injective (this function is called the continuum function).
It is worth mentioning Easton's theorem which tells us that we can pretty much modify the behavior of the continuum function as long as we preserve the basic properties of this function (for example, $\kappa<\lambda$ then $2^\kappa\leq 2^\lambda$).
This means that we can have the assertion that the continuum function is injective fail everywhere; we can have it failing unboundedly high; we can have it true unboundedly high AND fail unboundedly high as well...