Suppose I have an infinite dimension vector space V (not necessarily countably infinite). Suppose a have a set S that spans the space. If V is finite dimensional, it is trivial to construct a basis using elements in S that form a basis for V. Is this also valid when V has a non-countable number of dimensions?
EDIT: Wikipedia claims: "Theorem 3: Let V be a finite-dimensional vector space. Any set of vectors that spans V can be reduced to a basis for V by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set). If the axiom of choice holds, this is true without the assumption that V has finite dimension."
However, it doesn't provide a reference nor explain how the proof works.