I'm proving that $|X|\leq^*\mathbb{N}$ iff $\mathbb{N}\leq\mathcal{P}(X)$. One direction is trivial, but I'm not sure how to proceed for the other direction. Any suggestions?
Update
My apologies. The result as listed above is not necessarily correct, and in any case, not what I was trying to prove. I have made a new post with the relevant changes, here Given an injection $\mathbb{N}\to\mathcal{P}(X)$, how can we construct a surjection $X\to\mathbb{N}$?