Existence of a basis

Corollary (Existence of a Hamel basis). If $X$ is a vector space with $X\neq\{0\}$, then $X$ has a basis, i.e., a Hamel basis.

Connection to the extension theorem. Pick any nonzero $x\in X$. Then $\{x\}$ is linearly independent, so the extension theorem produces a basis containing $\{x\}$.