I wanted to show this to use it in some other proofs. I believe it holds. My questions are, if this proposition is true and if the proof sufficiently shows it holds.
Background
From a previous proof, it uses the result that $\mathcal{P}(\mathbb{Z_+})$, the power set of the positive integers, is uncountable.
Theorem 7.1 nonempty set A is countable $\iff\exists$ a surjective function $h: \mathbb{Z_+} \to A$ (citation: Munkres Topology Chapter 7)
Proof by Contradiction:
Let $A$ be a nonempty set and suppose $A$ is countable.
Assume $\exists$ a surjective function $h: A \to \mathcal{P}(\mathbb{Z_+})$. Since $A$ is countable, $\exists$ a surjective function $f: \mathbb{Z_+} \to A$, by Theorem 7.1.
Consider the composition $h\circ f: \mathbb{Z_+} \to \mathcal{P}(\mathbb{Z_+})$. Since $f$ and $h$ are surjections, it follows that $h\circ f$ is a surjection ($c\in \mathcal{P}(\mathbb{Z_+}) \Rightarrow \exists a\in A$, such that $c = h(a) \Rightarrow \exists n\in \mathbb{Z_+}$, such that $c = h(f(n))$.
This implies that $\mathcal{P}(\mathbb{Z_+})$ is countable, by Theorem 7.1, a contradiction. Hence, $A$ is uncountable.