Prove that if $v$ is a cut-vertex of a connected graph $G$, then $v$ is NOT a cut vertex of $G'$.
I know that a vertex $v$ is a cut vertex of $G$ IFF there exists vertices $u$ and $w$ $(u,w \not=v)$ such that $v$ is on every $u-w$ path of $G$.
I also know that the complement $G'$ of a graph $G$ is a graph with vertex set $V(G)$ such that two vertices are adjacent in $G'$ IFF they are NOT adjacent in $G$.
So is it correct to conclude that there are no $u-w$ paths in $G'$ on which $v$ is found?
I don't know how to set up this proof