Say that a set $\displaystyle A \subseteq \omega_1$ is stationary if it has nonempty intersection with every subset of $\displaystyle \omega_1$. Suppose that there is an injection from $\displaystyle \omega_1$ into $\displaystyle \mathcal{P}(\mathbb{N})$. Show that every stationary subset of $\displaystyle \omega_1$ can be split into two disjoint subsets of $\displaystyle \omega_1$.

Notation: $\displaystyle \omega_1$ denotes the first uncountable ordinal. We may us the Countable Principle of Choice, but not the Axiom of Choice.