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

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