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 Th22:
  X is epsilon-transitive implies
  for A st A <> {} holds Tarski-Class(X,A) is epsilon-transitive
proof
  assume
A1: Y in X implies Y c= X;
  defpred OnP[Ordinal] means
  $1 <> {} implies Tarski-Class(X,$1) is epsilon-transitive;
A2: for A st for B st B in A holds OnP[B] holds OnP[A]
  proof
    let A such that
A3: for B st B in A holds OnP[B] and
A4: A <> {};
    let Y such that
A5: Y in Tarski-Class(X,A);
A6: now
      given B such that
A7:   A = succ B;
A8:   B c= A by ORDINAL1:6,def 2,A7;
A9:  OnP[B] by A3,A7,ORDINAL1:6;
A10:  Tarski-Class(X,B) c= Tarski-Class(X,A) by A8,Th16;
  now
        assume not Y c= Tarski-Class(X,B);
        then consider Z such that
A11:    Z in Tarski-Class(X,B) and
A12:    Y c= Z or Y = bool Z by A5,A7,Th10;
A13:    Y = bool Z implies thesis by A7,A11,Th10;
    now
          assume
A14:      Y c= Z;
          thus thesis
          proof
            let x be object;
            reconsider xx=x as set by TARSKI:1;
            assume
A15:        x in Y;
then A16:        x in Z by A14;
A17:        now
              assume B = {};
then           Tarski-Class(X,B) = { X } by Lm1;
then           Z = X by A11,TARSKI:def 1;
              hence thesis by A7,A11,Th10,A1,A14,A15;
            end;
        now
              assume B <> {};
then           Z c= Tarski-Class(X,B) by A9,A11,ORDINAL1:def 2;
then           x in Tarski-Class(X,B) by A16;
              hence thesis by A10;
            end;
            hence thesis by A17;
          end;
        end;
        hence thesis by A12,A13;
      end;
      hence thesis by A10;
    end;
 now
      assume
A18:  for B holds A <> succ B;
then   A is limit_ordinal by ORDINAL1:29;
      then consider B such that
A19:  B in A and
A20:  Y in Tarski-Class(X,B) by A4,A5,Th13;
A21:  succ B <> A by A18;
A22:  Tarski-Class(X,B) c= Tarski-Class(X,succ B) by Th15;
A23:  succ B c< A by A19,ORDINAL1:21,A21;
A24:  Tarski-Class(X,succ B) c= Tarski-Class(X,A) by A19,ORDINAL1:21,Th16;
  Tarski-Class(X,succ B) is epsilon-transitive by A3,A23,ORDINAL1:11;
then   Y c= Tarski-Class(X,succ B) by A20,A22;
      hence thesis by A24;
    end;
    hence thesis by A6;
  end;
  thus for A holds OnP[A] from ORDINAL1:sch 2(A2);
end;
