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].

Lemma properEcard A B : (A \proper B) = (A \subset B) && (#|A| < #|B|).