Proving equivalent definitions for MSTs
In order to make sure $3)\implies1)\,$ is true, let us assume that all costs of edges are different. Otherwise, there is a counterexample at Is a set of acyclic |V| - 1 light edges always a Minimum Spanning Tree?.

Here is a hint to prove $3) \implies 1)\,$. Suppose you have some MST $T'$. Show that $T$ and $T'$ have the same number of edges.  Show that $T'$ must contain every minimal cut.