Prove that there is no convex polyhedron
Notice that the Euler theorem is not required. Since you found $V=4$, the polyhedron is necessarily a tetrahedron(and this can be proved easily by exhaustion). This procedure is more general, because it removes the convexity hyphothesis.
There are no polyhedron with 7 edges.
Even more generally you can prove easily that there exist polyhedron with $n$ edges with n≥6, with 7 being the only exception.