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

hereby :: thesis: ( CnIPC T = T implies T is IPC_theory )

assume A1: T is IPC_theory ; :: thesis: CnIPC T = T
A2: T c= CnIPC T by Th12;
CnIPC T c= T

proof

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in CnIPC T or a in T )
assume a in CnIPC T ; :: thesis: a in T
hence a in T by A1, Def15; :: thesis: verum

end;

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

end;

thus ( CnIPC T = T implies T is IPC_theory ) ; :: thesis: verum