Let $V$ be a vector space of dimension $dimV =n$, over a field F. $S$ is a linearly independent set. Suppose I am equipped with the fact that the size of any linearly independent set must be $\le$ the size of any spanning set, and with the fact that a basis is a minimal spanning set(with a proof akin to this one). Then, how can I prove that $\#S \le n$, and that if $\#S=n$ then $S$ is a basis? I wouldn't use any other theorems that aren't absolutely elementary beyond the two I am equipped with.
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