Trivial Vector Space iff Zero Dimension
Theorem
Let $V$ be a vector space.
Then $V = \set \bszero$ if and only if $\map \dim V = 0$, where $\dim$ signifies dimension of vector space.
Proof
Necessary Condition
Suppose $V = \set \bszero$.
We have that $V$ has no $\mathbf v \in V, \mathbf v \ne \bszero$.
Thus there exists no $\set {\mathbf v} \subseteq V$ such that $\mathbf v \ne \bszero$.
Such a $\set {\mathbf v}$ would be a linearly independent set.
Hence $\O$ is the only possible basis for $V$.
Hence $\map \dim V = \card \O = 0$.
$\Box$
Sufficient Condition
Suppose $\map \dim V = 0$.
Then by definition of dimension, $0 = \map \dim V = \card B$, where $B$ is a basis for $V$.
Thus $B = \O$.
Suppose there are sets $\set {\mathbf v}$ such that $\mathbf v \in V, \mathbf v \ne \bszero$.
By Singleton is Linearly Independent, any such $\set {\mathbf v}$ would be a linearly independent set.
So $V$ has no such $\mathbf v \ne \bszero$.
Thus $V = \set \bszero$.
$\blacksquare$