Is this a valid proof that every basis for a (not necessarily finite-dimensional) vector space $V$ has the same cardinality?
Independent sets have cardinality no greater than $dim(V)$ and spanning sets have cardinality at least $dim(V)$.
Let A and B be two bases for $V$. Since A spans and B is independent, $|A|\ge dim(V)\ge|B|$. Since B spans and A is independent, $|B|\ge dim(V)\ge|A|$. Therefore $|A| = |B|$.
But this may be faulty because it uses $dim(V)$ which is a meaningless concept before we have shown that all bases have the same cardinality. So, what is a correct approach... to assume there are two bases, and show there is a bijection between them?