Definition: The empty set E is a subset of S if every member of E is a member of S.
P (Proposition): E is a subset of S.
Proof: P is false if there are members of E which do not belong to S. There are no members of E which do not belong to S, so P is true.
Problem with proof: There are no members of E. So though I can talk about no members, I can’t talk about “no members which” because member doesn’t exist, which doesn’t refer to anything. The problem already occurs in the Definition: to say that “something that doesn’t exist is” is meaningless, it says that something that isn’t, is. E may exist, but its members do not.
Conclusion: E is not a subset of S