Let $\mathbb{A}$ be a non empty set and $V$ be a vector space on a field $\mathbb{K}$
Definition 1. An application $$f\colon\mathbb{A}\times\mathbb{A}\to V\quad (P,Q)\mapsto f(P,Q)$$ defines an affine space structure on $\mathbb{A}$ if the following properties are true
$(AF1)$ for all point $P$ of $\mathbb{A}$ and for all vector $\textbf{v}$ in $V$ exists a unique point $Q$ of $\mathbb{A}$ such that $f(P,Q)=\textbf{v}$;
$(AF2)$ for every triad $(P,Q,S)$ of point of $\mathbb{A}$ we have $$f(P,Q)+f(Q,S)=f(P,S).$$
Definition 2. A affine space on field $\mathbb{K}$ is a pair $$(\mathbb{A},f),$$ where $\mathbb{A}$ is a set, $V$ a vector space over $\mathbb{K}$ and $f\colon\mathbb{A}\times\mathbb{A}\to V$ defines an affine space structure on $\mathbb{A}$, as in definition 1.
Notation. $f(P,Q)$ will be denoted by $$\textbf{PQ}\quad\text{or}\quad\textbf{Q}-\textbf{P}.$$
Let $(\mathbb{A},f)$ be an affine space over a field $\mathbb{K}$ and let $V$ be a finite vector space over the same field. We define the dimension of $\mathbb{A}$ as $$\dim\mathbb{A}=\dim V.$$
Consider a point $O\in\mathbb{A}$ and we consider the application $$g_O\colon V\to \mathbb{A},\quad \textbf{v}\mapsto O+\textbf{v}.$$
Note. From definition $\textbf{v}+O$ is that point $Q$ in $\mathbb{A}$ such that $\textbf{OQ}=\textbf{Q}-\textbf{O}=\textbf{v}$
From $(AF2)$ follow that $g_O$ is a bijection, that induces a $\mathbb{K}-$ vector space structure on $\mathbb{A}$ which is isomorphic to $V$. We denote the affine space $\mathbb{A}$ with this vector space structure as $\mathbb{A}_O.$
Question. Why if $\dim(\mathbb{A})=0$ then $\mathbb{A}$ consists of a single point?
My solution. Let $O$ be a point of $\mathbb{A}$, then
$$\dim\mathbb{A}=0\implies \dim V=0\implies V=\textbf{0}\implies \mathbb{A}_O=\textbf{O}.$$
Since $\mathbb{A}_O=\mathbb{A}$ as affine space, (every vector space can be considered as an affine space) results that $\mathbb{A}$ consists of a single point. it's correct?