Connected Graph is Tree iff Removal of One Edge makes it Disconnected/Sufficient Condition/Proof 1

Theorem

Let $G = \struct {V, E}$ be a tree.

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

Proof

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 $G$.

$\blacksquare$