That's not correct. For example, take this graph:
This graph has three vertices and one edge.
Your statement "Ex Ey: ~edge(x, y)" is true for this graph -- for example, I can select x=A and y=B, and then "~edge(A, B)" is a true statement. However, the graph obviously does not have zero edges!
In English, your statement says: "There exists a pair of vertices that don't have an edge between them". Do you see why?