safe edge theorem proof clarification

Question

I found the following proof for the theorem that states "A light edge that crosses a cut that respects A is safe for A":

See: https://www2.hawaii.edu/~janst/311_f19/Notes/Topic-17.html where also all the necessary definitions are given.

What I do not understand in this proof is that we have here T that is a MST and then we say that T' is also a MST that contains the edge u-v and not x-y. But if the weight of u-v is less than that of x-y ("w(T) - w(x,y) + w(u,v) ≤ w(T)"), then T could have never been a MST (because it would have never chosen x-y over u-v)? Does someone see what goes wrong in my explanation?

Answer 1

Although correct, that statement, "$w(T') = w(T) - w(x,y) + w(u,v) \le w(T)$" alone is indeed confusing, since it leaves the impression that $w(T')$ might be smaller than $w(T)$.

Well, the equal part of "$\le$" comes to rescue. It is possible that $w(T')=w(T)$.

Indeed, that must be the case, since we also have $w(T)\le w(T')$ by the definition of MST and $T$ being an MST.

There is nothing wrong to say "$w(T')\le w(T)$", just as we say "$3\le 3$".

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Furthermore, we know that $w(x,y)=w(u,v)$. This is the fact that the edge in any MST that crosses a cut must be an edge of minimum weight that crosses that cut. (All minimum edges crossing the same cut have the same weight, of course.)