An D21: Algebraic structure $G = (X, +)$ is an abelian group if and only if

(1)	\begin{equation} \forall \, x, y \in X : x + y \in X \end{equation}
(2)	\begin{equation} \forall \, x, y, z \in X : (x + y) + z = x + (y + z) \end{equation}
(3)	\begin{equation} \exists \, 0_G \in X : \forall \, x \in X : 0_G + x = x + 0_G = x \end{equation}
(4)	\begin{equation} \forall \, x \in X : \exists \, {-} x \in X : -x + x = x + (- x) = 0_G \end{equation}
(5)	\begin{equation} \forall \, x, y \in X : x + y = y + x \end{equation}