reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;

theorem Th18:
  Tarski-Class(X,A) = Tarski-Class(X,succ A) implies
  Tarski-Class(X,A) = Tarski-Class X
proof
  assume
A1: Tarski-Class(X,A) = Tarski-Class(X,succ A);
 {} c= A;
then A2: Tarski-Class(X,{}) c= Tarski-Class(X,A) by Th16;
A3: Tarski-Class(X,{}) = { X } & X in { X } by Lm1,TARSKI:def 1;
 Tarski-Class(X,A) is_Tarski-Class_of X
  proof
    thus X in Tarski-Class(X,A) by A2,A3;
A4: Tarski-Class(X,succ A) = { u : ex v st v in Tarski-Class(X,A) & u c= v } \/
    { bool v : v in Tarski-Class(X,A) } \/
    bool Tarski-Class(X,A) /\ Tarski-Class X by Lm1;
 Tarski-Class X is_Tarski-Class_of X by Def4;
then A5: Tarski-Class X is Tarski;
    thus for Z,Y being set st
    Z in Tarski-Class(X,A) & Y c= Z holds Y in Tarski-Class(X,A)
    proof
      let Z, Y be set;
      assume
A6:   Z in Tarski-Class(X,A) & Y c= Z;
  Tarski-Class X is_Tarski-Class_of X by Def4;
then   Tarski-Class X is Tarski;
then   Tarski-Class X is subset-closed;
      then reconsider y = Y as Element of Tarski-Class X by A6;
  ex v st v in Tarski-Class(X,A) & y c= v by A6;
then   Y in { u : ex v st v in Tarski-Class(X,A) & u c= v };
then   Y in { u : ex v st v in Tarski-Class(X,A) & u c= v } \/
      { bool v : v in Tarski-Class(X,A) } by XBOOLE_0:def 3;
      hence thesis by A1,A4,XBOOLE_0:def 3;
    end;
    thus Y in Tarski-Class(X,A) implies bool Y in Tarski-Class(X,A)
    proof
      assume Y in Tarski-Class(X,A);
then   bool Y in { bool u : u in Tarski-Class(X,A) };
then   bool Y in { u : ex v st v in Tarski-Class(X,A) & u c= v } \/
      { bool v : v in Tarski-Class(X,A) } by XBOOLE_0:def 3;
      hence thesis by A1,A4,XBOOLE_0:def 3;
    end;
    let Y;
    assume that
A7: Y c= Tarski-Class(X,A) and
A8: not Y,Tarski-Class(X,A) are_equipotent;
 Y c= Tarski-Class X by A7,XBOOLE_1:1;
then
 Y,Tarski-Class X are_equipotent or Y in Tarski-Class X by A5;
    hence thesis by A1,A7,A8,Th10,CARD_1:24;
  end;
  hence Tarski-Class(X,A) c= Tarski-Class X &
  Tarski-Class X c= Tarski-Class(X,A) by Def4;
end;
