Find a diffrent minimal spanning tree for a graph For my homework I have a problem that I can't solve and it makes me wonder about 2 different MST:

Let $G=(V,E)$ be a graph that has a minimum spanning tree $T$.
I want to find another minimum spanning tree $T'$ that has at least 1 different edge $e'$
such that the weight of $e'$ is differ from any weight of edges in $T$.

If $T'$ doesn't exist I can claim that every 2 different MST must have the same weight for each edge.
My intuition says that this claim is wrong but on the other hand I can't find example of $T'$ to contradict this claim.
You cannot find such a MST. Every two minimal spanning trees must have the same multiset of edge-weights.
Actually you can find the proof in the link that Raphael added:

Do the minimum spanning trees of a weighted graph have the same number of edges with a given weight?