reserve X,Y,Z,X1,X2,X3,X4,X5,X6 for set, x,y for object;
reserve a,b,c for object, X,Y,Z,x,y,z for set;
reserve A,B,C,D for Ordinal;

theorem
  for X ex A st not A in X & for B st not B in X holds A c= B
proof
  let X;
  defpred P[object] means not $1 in X;
  consider B such that
A1: not B in X by Th22;
  consider Y such that
A2: for a being object holds a in Y iff a in succ B & P[a]
   from XBOOLE_0:sch 1;
  for a be object holds a in Y implies a in succ B by A2;
  then
A3: Y c= succ B;
  B in succ B by Th2;
  then Y <> {} by A1,A2;
  then consider A such that
A4: A in Y and
A5: for B st B in Y holds A c= B by A3,Th16;
  A in succ B by A2,A4;
  then
A6: A c= succ B by Def2;
  take A;
  thus not A in X by A2,A4;
  let C;
  assume
A7: not C in X;
  assume
A8: not A c= C;
  then not A in C by Def2;
  then C in A by A8,Th10;
  then C in Y by A2,A7,A6;
  hence contradiction by A5,A8;
end;
