Well-ordering theorem implies the axiom of choice

Let \(\mathcal A\) be a collection of nonempty sets. Put \[ U=\bigcup_{A\in\mathcal A}A. \] By the well-ordering theorem, choose a well-order \(<\) on \(U\). For each \(A\in\mathcal A\), the subset \(A\) is nonempty, so it has a unique \(<\)-smallest element. Define \[ c(A)=\min_{<}A. \] Then \(c(A)\in A\) for every \(A\in\mathcal A\). Thus \(c\) is a choice function on \(\mathcal A\), proving the axiom of choice.