(** We assume a type with two elements. They play the role of booleans. The main theorem under the current assumptions is that [T=F] *)
Variable Bool : Prop.
Variable T : Bool.
Variable F : Bool.
(** The powerset operator *)
Definition pow (P:Prop) := P -> Bool.
(** A piece of theory about retracts *)
Section Retracts.
Variables A B : Prop.
Record retract : Prop :=
{i : A -> B; j : B -> A; inv : forall a:A, j (i a) = a}.
Record retract_cond : Prop :=
{i2 : A -> B; j2 : B -> A; inv2 : retract -> forall a:A, j2 (i2 a) = a}.
(** The dependent elimination above implies the axiom of choice: *)

Lemma AC : forall r:retract_cond, retract -> forall a:A, j2 r (i2 r a) = a.
Definition U := forall P:Prop, pow P.