let T be Subset of MC-wff ; :: thesis: ( T is CPC_theory iff CnCPC T = T )

hereby :: thesis: ( CnCPC T = T implies T is CPC_theory )

assume A1: T is CPC_theory ; :: thesis: CnCPC T = T
A2: T c= CnCPC T by Th72;
CnCPC T c= T

proof

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in CnCPC T or a in T )
assume a in CnCPC T ; :: thesis: a in T
hence a in T by A1, Def20; :: thesis: verum

end;

hence CnCPC T = T by A2, XBOOLE_0:def 10; :: thesis: verum

end;

thus ( CnCPC T = T implies T is CPC_theory ) ; :: thesis: verum