The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times, as well as having applications for geometry and topology. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes at the cost of potential fattening, which reflects genuine non-uniqueness of the flow after singular times. Even for flows starting from smooth, embedded, closed initial data, such non-uniqueness can occur. Thus, we can't expect genuine uniqueness in general. Addressing this non-uniqueness issue is a difficult problem. With Alec Payne, we establish an intersection principle comparing two intersecting flows. We prove that level set flows satisfy this principle in the absence of non-uniqueness.