In Appendix A of chapter 2 of The Quantum Theory of Fields, vol. 1 by Weinberg, he presents a proof of Wigner's theorem: given a symmetry transformation $T$ of rays, one can extend this to a symmetry transformation (which must be either unitary and linear, or antiunitary and antilinear) on the full Hilbert space.
Quoting directly from page 91:
To start, consider some complete orthonormal set of state-vectors $\Psi_k$ belonging to rays $\mathscr{R}_k$, with ($\Psi_k,\Psi_l) = \delta_{kl}$, and let $\Psi'_k$ be some arbitrary choice of state-vectors belonging to the transformed rays $T\ \mathscr{R}_k$.
My questions:
(1) Am I correct in thinking that Weinberg's proof uses the axiom of choice (to show the existence of the $\Psi'_k$)?
(2) If so, can Weinberg's argument be modified so as not to use the axiom of choice?