I suppose this problem should be a commonplace, but I only find this one, in which notations are kind of idiosyncratic, along with a glaring defect.
My question is where I can find a more formal proof?
I will assume choice, otherwise things get extra messy.
Let $X$ be of cardinality $\kappa$.
We have, if so, $2^\kappa$ partitions of $X$ whenever $|X|=\kappa$.