[a set/sets] [an element/elements of x]
[x is in/from y @ x is an element of y]
[x belongs/belong to y @ x is in y]
[a subset/subsets of x] [x is empty]

Definition DefSubset. Let S be a set.
  A subset of S is a set with no elements not in S.

Definition DefEmpty. Let S be a set.
  S is empty iff S has no elements.

Axiom ExEmpty. There exists an empty set.

Proposition. Let S be a set.
  S is a subset of every set iff S is empty.
Proof.
  First assume S is a subset of every set.
    Then S is empty.
    Indeed
      Let z be in S and E be an empty set.
      Then z is an element of E.
      We have a contradiction.
    end.
  end.
  Now assume S is empty.
    Let T be a set.
    Every element of S is in T (by DefEmpty).
    Hence S is a subset of T (by DefSubset).
  end.
qed.

