Connected Graph is Tree iff Removal of One Edge makes it Disconnected

Theorem
Let $G = \struct {V, E}$ be a connected simple graph.

Then $G$ is a tree :
 * for all edges $e$ of $G$, the edge deletion $G \setminus \set e$ is disconnected.

Sufficient Condition
Let $G$ be a tree.

Then by definition $G$ has no circuits.

From Condition for Edge to be Bridge, every edge of $G$ is a bridge.

Thus by definition of bridge, removing any edge of $G$ will disconnect $G$.

Necessary Condition
Let $G$ be a connected simple graph such that for all edges $e$ of $G$, the edge deletion $G \setminus \set e$ is disconnected.

Hence, by definition, every edge of $G$ must be a bridge.

So by Condition for Edge to be Bridge, $G$ has no circuits.

Hence $G$ is a tree by definition.