Hello, I have enrolled in my college's "Design and Analysis of Algorithms" course. Recently we were studying Greedy algorithms and proving their correctness. I have a doubt about the correctness proof of Prim's Algorithm. How do we prove that it does not form a cycle during its execution? In the following snippet from our lecture slides, it says:

"No cycles ever get created in T. Why? Consider any iteration with current sets X and T. Suppose e gets added. Key point: e is the first edge crossing (X, V — X) that gets added to T -> its addition can't create a cycle in T (by Lonely Cut Corollary). QED!"