reserve V, C, x, a, b for set;
reserve A, B for Element of SubstitutionSet (V, C);
reserve C for finite set;
reserve A, B for Element of SubstitutionSet (V, C);

theorem Th10:
  A = {} implies -A = { {} }
proof
  defpred P[Element of PFuncs (Involved A, C)] means for g be Element of
  PFuncs (V, C) st g in A holds not $1 tolerates g;
  assume
A1: A = {};
  then
A2: for g be Element of PFuncs (V, C) st g in A holds not {} tolerates g;
  { xx where xx is Element of PFuncs (Involved A, C) : P[xx]} c= PFuncs (
  Involved A, C) from FRAENKEL:sch 10;
  then -A c= PFuncs ({}, C) by A1,Th7;
  then
A3: -A c= { {} } by PARTFUN1:48;
  {} in { {} } by TARSKI:def 1;
  then {} in PFuncs ({}, C) by PARTFUN1:48;
  then {} in PFuncs (Involved A, C) by A1,Th7;
  then {} in -A by A2;
  then { {} } c= -A by ZFMISC_1:31;
  hence thesis by A3;
end;
