reserve m for Cardinal,
  A,B,C for Ordinal,
  x,y,z,X,Y,Z,W for set,
  f for Function;
reserve f,g for Function,
  L for Sequence,
  F for Cardinal-Function;
reserve U1,U2,U for Universe;
reserve u,v for Element of U;
reserve u,v for Element of Tarski-Class(X);

theorem
  Tarski-Class(X,{}) in Tarski-Class(X,1) &
  Tarski-Class(X,{}) <> Tarski-Class(X,1)
proof
  deffunc F(Ordinal) = Tarski-Class(X,$1);
  deffunc C(Ordinal,set) = { u : ex v st v in $2 & u c= v } \/
  { bool v : v in $2 } \/ bool $2 /\ Tarski-Class X;
  deffunc D(Ordinal,Sequence) = (union rng $2) /\ Tarski-Class X;
A1: for A for x being object holds x = F(A) iff
  ex L st x = last L & dom L = succ A & L.0 = { X } &
  (for C st succ C in succ A holds L.succ C = C(C,L.C)) &
  for C st C in succ A & C <> 0 & C is limit_ordinal
  holds L.C = D(C,L|C) by CLASSES1:def 5;
A2: F(0) = { X } from ORDINAL2:sch 8(A1);
  then X in Tarski-Class(X,{}) by TARSKI:def 1;
  then
A3: bool X in Tarski-Class X by CLASSES1:4;
 { X } c= bool X
  proof
    let x be object;
    assume x in { X };
then  x = X by TARSKI:def 1;
    hence thesis by ZFMISC_1:def 1;
  end;
then  1 = succ 0 & { X } in Tarski-Class X by A3,CLASSES1:3;
  thus
then A4: Tarski-Class(X,{}) in Tarski-Class(X,1) by A2,CLASSES1:10;
  assume Tarski-Class(X,{}) = Tarski-Class(X,1);
  hence contradiction by A4;
end;
