Let $(A_x)_{x\in X}$ be a collection of disjoint subsets of $\mathbb{N}$. Using the Axiom of Choice, i.e. assuming there exists a function $f:\{A_x\}_{x\in X}\to \bigcup_{x\in X} A_x$, with $f(A_x)\in A_x$ for all $x\in X$, it follows by disjointness that this collection injects into $\mathbb{N}$.
My question is this: Are there proofs of this result that are not reliant on choice? Maybe DC? P.S. this is not my area of expertise :P Thanks!!