I am reviewing Diestel's Graph Theory and we are asked to prove that the following are equivalent:
(i) $T$ is a tree.
(ii) Any two vertices of $T$ are linked by a unique path in $T$.
(iii) $T$ is connected but $T-e$ is disconnected for all $e\in E(T)$.
(iv) $T$ contains no cycle but $T+xy$ does for any two nonadjacent vertices $x,y\in T$.
The definition of a tree we start with is that it is a connected acyclic graph. I would like to show $(i)\implies(ii)\implies(iii)\implies(iv)\implies(i)$ to finish this.
I have shown $(i)\implies(ii)$ and $(ii)\implies (iii)$ and am stuck on $(iii)\implies(iv)$. On the one hand since $T$ is connected then for any nonadjacent vertices $x$ and $y$, $T+xy$ will contain a cycle: the $xy-$path in $T$ along with the edge $xy$. This seems to prove the claim, but I do not use the idea that $T-e$ is disconnected for every $e\in E(T)$. Am I missing something?
If $T$ contains a cycle, then removing an edge $e$ from this cycle leaves a graph $T-e$ which is still connected, contradicting (iii) (which says that every edge of $T$ is a bridge).