How to prove a function is well defined

Question

Let B be the set of all infinite cardinals b such that $b^2 = b$.

I need to show that this function:

$$λb ∈ B.ιx (x^b = 2^b) and (∀c.(c>x) → (c^b > 2^b))$$

is well defined. It seems like everything I try doesnt hold $c^b > 2^b$

Ideas?

Thanks.