Section BasicSetTheory.
Variable T : finType.
Implicit Types (x : T) (A B : {set T}) (pA : pred T).
Canonical set_of_subType := Eval hnf in [subType of {set T}].
Canonical set_of_eqType := Eval hnf in [eqType of {set T}].
Canonical set_of_choiceType := Eval hnf in [choiceType of {set T}].
Canonical set_of_countType := Eval hnf in [countType of {set T}].
Canonical set_of_subCountType := Eval hnf in [subCountType of {set T}].
Canonical set_of_finType := Eval hnf in [finType of {set T}].
Canonical set_of_subFinType := Eval hnf in [subFinType of {set T}].
Definition set0 := [set x : T | false].
Definition setTfor (phT : phant T) := [set x : T | true].
(* other inclusions *)
Arguments setIidPl {A B}.

Lemma imset2S (A B : {pred aT}) (A2 B2 : {pred aT2}) :
A \subset B ->  A2 \subset B2 -> f2 @2: (A, A2) \subset f2 @2: (B, B2).