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+Definition relation (A : Type) := A -> A -> Type.
+Class Reflexive {A} (R : relation A) := reflexivity : forall x : A, R x x.
+Axiom IsHProp : Type -> Type.
+Existing Class IsHProp.
+Inductive Empty : Set := .
+Notation "~ x" := (x -> Empty) : type_scope.
+Record hProp := BuildhProp { type :> Type ; trunc : IsHProp type }.
+Arguments BuildhProp _ {_}.
+Canonical Structure default_hProp := fun T P => (@BuildhProp T P).
+Generalizable Variables A B f g e n.
+Axiom trunc_forall : forall `{P : A -> Type}, IsHProp (forall a, P a).
+Existing Instance trunc_forall.
+Inductive V : Type := | set {A : Type} (f : A -> V) : V.
+Axiom mem : V -> V -> hProp.
+Axiom mem_induction
+: forall (C : V -> hProp), (forall v, (forall x, mem x v -> C x) -> C v) -> forall v, C v.
+Definition irreflexive_mem : forall x, (fun x y => ~ mem x y) x x.
+Proof.
+  pose (fun x => BuildhProp (~ mem x x)).
+  refine (mem_induction (fun x => BuildhProp (~ mem x x)) _); simpl in *.
+  admit.