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 Th17:
  A in B iff succ A c= B
proof
  thus A in B implies succ A c= B
  proof
    assume
A1: A in B;
    then for a be object holds a in { A } implies a in B by TARSKI:def 1;
    then
A2: { A } c= B;
    A c= B by A1,Def2;
    hence thesis by A2,XBOOLE_1:8;
  end;
  assume
A3: succ A c= B;
  A in succ A by Th2;
  hence thesis by A3;
end;
