Suppose that using Zorn's lemma, we have proven that an object with some properties exists and then we've proven that such object is unique. Can we always conclude that we can prove the existence (and uniqueness) of the object without using Zorn's lemma (or any other equivalent form of axiom of choice)?
I was studying differentiable manifolds and I arrived to a point where it was proven that any differentiable atlas is contained in some unique maximal differentiable atlas. Then I found out that it can be proven without using axiom of choice. But isn't that because the maximal atlas is unique?
Everywhere I remember that it was necessary to use axiom of choice for proving the existence of some object, it was later proven that such object is not unique and in fact, there are uncountably many of such objects. So I asked the above question, but couldn't find any good answer.