Will the minimum spanning tree not have only those edges specified by the cycle property?
Sorry if I didn't understand your question, but what I think you meant to ask is, whether it is possible for an edge to not be in a MST if this edge is not in a cycle.

Here's my answer.

Let us assume that we have an edge (u,v) in a graph G, that isn't part of a cycle. This would mean that there is no path from u to v that isn't the edge (u,v).
If the edge (u,v) weren't in the MST, this would mean that there is no path from u to v in this MST, meaning that the MST would be disconnected, and thus not an MST at all.
Hence if an edge is not part of a cycle, it must be in the MST of the graph.