Edge is Bridge iff in All Spanning Trees

Theorem
Let $G$ be a simple graph.

Let $e$ be an edge of $G$.

Then $e$ is a bridge in $G$ $e$ belongs to every spanning tree for $G$.

Necessary Condition
Let $e$ be a bridge.

Then by definition there exist vertices $u$ and $v$ of $G$ such that $e$ belongs to every path between $u$ and $v$.

Let $T$ be an arbitrary spanning tree for $G$.

By definition $T$ is a connected subgraph of $G$ which contains $u$ and $v$.

So $u$ and $v$ are connected in $T$.

Therefore $e$ is in $T$.