Minimum spanning tree with two minimum edge weights

Given an undirected weighted graph $G$ with two edges of minimum weight and all other edges are distinct. Does G have a unique minimum spanning tree?

I know the proof for if all edge weights are distinct (it does give a unique MST) and I am thinking that if two edges of minimum weight are in $G$ then I should be able to show a counter example. But so far I have not been able to produce one.

So my question is does it give a unique MST if the graph $G$ contains two minimum weight edges?

Yes $G$ has a unique MST (assuming it is not a multigraph). Take a look at Kruskal's algorithm: the edges are ordered ascending by weight and then added to the MST unless there is a cycle. After adding the two minimum edges, you can't have a cycle, therefore both are in the MST.

Edit: As Paresh pointed out, this is not a complete proof, since there may be MSTs which cannot be produced by Kruskal's after permutating equally weighted edges in the sorted list. But Kruskal's is in fact able to produce any MST. Proof.

StackExchange Computer Science Q#7414, answer score: 5