(* P is a collection of a *)
(* P is a sub-collection of Q *)
Parameter MD4                : forall P Q, P ε subcoll Q ≡≡ (P ε P ∧ forall C, (C ε el P ⊃ C ε el Q)).
(* P is a external to Q *)
Parameter MD5                : forall P Q, P ε ext Q ≡≡ (P ε P ∧ forall C, ¬(C ε el P ∧ C ε el Q)).
(* P is a complement of Q relative to collection R *)
Parameter MD6                : forall P Q R, P ε compl Q R ≡≡ (P ε P ∧ Q ε subcoll R ∧ P ε (Kl (meet (el R)(ext Q)))).
(* P and Q overlap *)
Parameter MD7                : forall P Q, P ε ov Q ≡≡ (P ε P ∧ ¬forall C, ¬(C ε el P ∧ C ε el Q)).
(* The a's are discrete  *)
Parameter MD8                : forall a, discrete a ≡≡ forall P Q, (P ε a ∧ Q ε a) ⊃ singular_equality P Q ∨ P ε ext Q. (*MD8*)
(* A is the universe *)
Parameter MD9                : forall A, A ε U ≡≡ (A ε Kl universal).
(* A belongs to the union of a and b *)
Parameter MD10               : forall A a b, A ε (union a b) ≡≡ (A ε A ∧ (A ε a ∨ A ε b)).
(* =========================== Eliminations for complex existentials  ========================== *)

(* Specifications: *)
Lemma elim_ex2 : forall a B:N, ¬(forall C D, ¬(C ε a ∧ D ε el C ∧ D ε el B)) ⊃
exists C D, (C ε a ∧ D ε el C ∧ D ε el B).
Lemma intro_ex2 : forall a B:N, (exists C D, (C ε a ∧ D ε el C ∧ D ε el B)) ⊃
¬(forall C D, ¬(C ε a ∧ D ε el C ∧ D ε el B)).
Lemma intro_ex_coll1 : forall a A:N, (exists C:N, (C ε a ∧ C ε el A)) ⊃ ¬(forall C, ¬(C ε a ∧ C ε el A)).
Lemma intro_ex_3 : forall A B:N, (exists a:N, (B ε (Kl a) ∧ A ε a)) ⊃ ¬(forall a, ¬(B ε (Kl a) ∧ A ε a)).
Lemma elim_ex_3 : forall A B:N, ¬(forall a, ¬(B ε Kl a ∧ A ε a)) ⊃ exists a:N, (B ε Kl a ∧ A ε a).
Lemma elim_ex_coll2 : forall a P Q:N, ¬(forall C D, ¬(C ε a ∧ C ε el P ∧ D ε el C ∧ D ε el Q)) ⊃
exists C D, (C ε a ∧ C ε el P ∧ D ε el C ∧ D ε el Q).
Lemma intro_ex_coll2 : forall a P Q:N, (exists C D, (C ε a ∧ C ε el P ∧ D ε el C ∧ D ε el Q)) ⊃
¬(forall C D, ¬(C ε a ∧ C ε el P ∧ D ε el C ∧ D ε el Q)).