Minimum Spanning Tree: What exactly is the Cut Property?

A cut of a connected graph is a minimal set of edges whose removal separate the graph into two components (pieces). The minimal cut property says that if one of the edges of the cut has weight smaller than any other edge in the cut then it is in the MST. To see this, assume that there is an MST not containing the edge. If we add the edge to the MST we get a cycle that crosses the cut at least twice, so we can break the cycle by removing the other edge from the MST, thereby making a new tree with smaller weight, thereby contradicting the minimality of the MST.

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