Vector Space has Basis

Theorem
Let $V$ be a vector space.

Then $V$ has a basis.

Proof
The result follows from Vector Space has Basis:

By Singleton is Linearly Independent, $L$ can be taken to be any singleton of $V$.

$S$ can be taken to be $V$, since $V$ trivially spans itself.

Therefore, $L$ and $S$ exist, so $V$ has a basis.