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.

There's another property up on which this explanation is based.

"For any of the cut, if there are even number of edges crossing the cut then there must be a cycle crossing the cut"

Because MST does not contain any cycle there won't be any even number of edges crossing the cut.

Proof by contradiction: Assume that there's a MST not containing edge with min weight "e". If we add the edge "e" to the MST we get a cycle crossing the cut at least twice. We can remove other edge with more weight and break the cycle which results in an ST containing lesser weighing edge "e". This is in contradiction with the assumption.