The "injective continuum function hypothesis" (ICF) is the following statement.

ICF (Version 0). For all cardinal numbers $\kappa$ and $\nu$, we have $2^\kappa = 2^\nu \rightarrow \kappa = \nu.$

I really like this axiom, because its equivalent to:

ICF (Version 1). For all sets $X$ and $Y$, we have:

$$|\mathcal{P}(X)| = |\mathcal{P}(Y)| \rightarrow \mathcal{P}(X) \cong \mathcal{P}(Y)$$

(On the left we have equality of cardinal numbers; on the right, isomorphicness of posets.)

This means that, under ICF, cardinality is "more useful" than it might otherwise be, because all we need to know about $\mathcal{P}(X)$ and $\mathcal{P}(Y)$ to tell whether or not they're isomorphic as posets is their cardinality.

Question. What other potential axioms for set theory can be written in the form: "If mathematical structures $X$ and $Y$ are equipotent, then they're isomorphic"?