Proof required for edges in a minimum spanning tree G=(V,E) is an undirected simple graph in which each edge e has a distinct weight, and e is a particular edge of G. Which of the following statements about the minimum spanning trees (MSTs) of G is/are TRUE?

I. If e is the lightest edge of some cycle in G, then every MST of G includes e.

II. If e is the heaviest edge of some cycle in G, then every MST of G excludes e.

The answer is only statement 2 is correct.

I was able to prove statement 1 is false using contradiction. When we think of a complete graph with 4 vertices and edge weights 1,2,5,6 in non diagonal and diagonal edges 3 and 4. 4,5,6 will create a cycle and we can exclude the lighest edge e (4) from it, in a MST.

But statement 2 cannot be proved using the same. What should be the proof for proving statement 2 is correct?
Hint: Assume the contrary, that is, there exists at least one MST that contains the heaviest edge of some cycle in G. What happens if you remove that edge from MST? May it be the case that another edge in the cycle can play the 'role' of e in MST?