Algebraic system $\mathfrak A$ contains a proper subsystem $\mathfrak B$ isomorphic to $\mathfrak A$. Prove that $\mathfrak A$ is contained in a proper supersystem $\mathfrak C$ isomorphic to $\mathfrak A$.
I know how to prove this using Tarski's theorem about choice (and, hence, axiom of choice).
The question is: Could this theorem be proved independently of axiom of choice? And, if so, how?