Trivial Vector Space iff Zero Dimension

Theorem
Let $V$ be a vector space.

Then $V = \left\{{\mathbf 0}\right\}$ iff $\dim \left({V}\right) = 0$, where $\dim$ signifies dimension.

Necessary Condition
Suppose $V = \left\{{\mathbf 0}\right\}$.

As $V$ has no linearly independent vectors, $\varnothing$ is the only possible basis for $V$,

Hence, $\dim \left({V}\right) = \left|{\varnothing}\right| = 0$.

Sufficient Condition
Suppose $\dim \left({V}\right) = 0$.

Then by definition of dimension, $0 = \dim \left({V}\right) = \left\vert|{B}\right\vert|$, where $B$ is a basis for $V$.

Hence $B = \varnothing$, and so $V$ has no linearly independent vectors; thus $V = \left\{{\mathbf 0}\right\}$.