Let $H$ be a connected graph and it contains two spanning trees $A, B$. $A, B$ do not share any edges with each other (H may contain edges in neither A or B). Show $H$ does not have a bridge.
For contradiction suppose edge $e$ is a bridge and in $H$ and let $e = uv$ so Now suppose there are 2 connected components of $H - e$: $G_1, G_2$ such that $u$ is in $G_1$ and $v$ is in $G_2$.
However, notice that if $e$ is not in $A$ or if $e$ is not in $B$ then it is not possible for either of the trees to cover one of $\{u, v\}$ vertices so $e$ must be in both $A, B$. Contradiction.
Is this correct justification?