Let $(G,\cdot)$ be a group. We define a relation on $G$ as follows: if $a,b\in G$ we write $a\sim b$ to mean that there exists $g\in G$ such that $ga=bg$. Let $x\in G$. Prove that if $[x]=\{x\}$ then $x$ commutes with every element of $G$.
Notes:
$\cdot$The relation is an equivalence relation, fulfilling the following conditions:
1)$\forall x, x\sim x$
2)$\forall x\forall y$,$x\sim y \Rightarrow y \sim x$
3)$\forall x \forall y \forall z$, $(x \sim y )\wedge (y \sim z)\Rightarrow x \sim z$
$\cdot$ $[x]$ denotes the equivalence class of $x$, which is the set of all elements $y$ in the domain for which $x \sim y$