Spanning Tree - Equivalent Properties
Continuing where @ijkilchenko left off.

$2\to 3$: Let $T$ have the cut property and let $T'$ be any other spanning tree of $G$. For each $e\in T$, there is some edge $e'\in T'$ that crosses the cut defined by $e$. Define $f : E(T) \to E(T')$ by mapping each edge $e \in E(T)$ to the edge $e' \in E(T)'$ with smallest weight among those crossing the cut defined by $e$.  According to the cut property $w(e)\le w(f(e))$, so we get 
$$ w(T) = \sum_{e \in T} w(e) \le \sum_{e\in T} w(f(e))\le \sum_{e'\in T'} w(e') = w(T')$$
where the third inequality comes from the fact that if $f$ is not injective, it only decreases weights.  Since $T'$ was arbitrary, we have shown that $w(T)$ is minimum among all spanning trees of $G$, and we are done.

$3\to 1$: If $T$ does not have the cycle property, the definition supplies us with an edge $e\in T$ and $e'\notin T$ such that $w(e') &lt; w(e)$, so $T - e + e'$ is a tree with weight less than $T$. Therefore $T$ does not have minimum weight.