Number of Components after Removal of Bridge

Theorem
Let $$G = \left({V, E}\right)$$ be a graph.

Let $$e \in E$$ be a bridge.

Let $$m$$ be the number of components of $$G$$.

Then when $$e$$ is removed from $$G$$, the number of components in the remaining graph is $$m+1$$.

Proof
It is clear that, by definition of a bridge that removing $$e$$ increases the number of components.

So after $$e$$ is removed from $$G$$, the number of components in the remaining graph is at least $$m+1$$.

Suppose that removing $$e$$ disconnects $$G$$ into more than $$m+1$$ components.

Since $$e$$ joins only two vertices of $$G$$, it can link at most two of these components.

So there is at least one extra component when $$e$$ is put back into $$G$$, and so $$G$$ has more than $$m$$ components.

This contradicts the fact that $$G$$ has $$m$$ components.

Hence the result.