Let be an undirected graph with an edge weight . Suppose is part of an MST. Let be a cut where no edge of is in the cut edges. Then for of minimum weight - is part of an MST.
Proof
If then and we are done.
Assume .
Let . As is an MST it contains a path from to . As and the path contains an edge .
Let .
We have the size of is
as is a tree.
Consider the cycle - rewrite this cycle as . We will use this to show is connected.
Let . As is connected there must a path in connecting to .
If uses replace by to form which only uses edges in . As connects to we have is connected.
As is connected on and has edges it has to be a spanningtree of .
Note as is the minimum weight edge in we have and moreover
This gives that is of minimum weight as was a MST.
All together this gives us that is an MST where , proving the statement.