Trivial Vector Space iff Zero Dimension

Theorem
Let $V$ be a vector space

Then $V=\{0\}$ iff $\dim(V)=0$

Proof
If $V=\{0\}$, then by definition of basis; $\emptyset$ is the only posible basis in $V$; since $V$ has no linearly independent vectors. Thus $\dim(V)=|\emptyset|=0$

If $\dim(V)=0$, then by definition of dimension $0=\dim(V)=|B|$ where $B$ is a basis for $V$.

Hence $B=\emptyset$ and $V$ has no linear independent vectors; thus $V=\{0\}$