On wikipedia, there is a proof for the cycle property of the Minimum Spanning Tree as follows:
Cycle Property: For any cycle C in the graph, if the weight of an edge e of C is larger than the weights of all other edges of C, then this edge cannot belong to an MST.
Proof: Assume the contrary, i.e. that e belongs to an MST T1. Then deleting e will break T1 into two subtrees with the two ends of e in different subtrees. The remainder of C reconnects the subtrees, hence there is an edge f of C with ends in different subtrees, i.e., it reconnects the subtrees into a tree T2 with weight less than that of T1, because the weight of f is less than the weight of e.
Why does the proof assume that e belongs to MST? If e belongs to a cycle, and it belongs to MST, doesn't it immediately imply that the MST has a cycle and therefore is not a tree?
Please show me why I am not understanding this proof correctly. i.e. How can we have an edge e already in the tree but the tree remains acyclic?