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 Th13:
  x is Ordinal implies succ x is Ordinal
proof
A1: now
    let y;
A2: y in { x } implies y = x by TARSKI:def 1;
    assume y in succ x;
    hence y in x or y = x by A2,XBOOLE_0:def 3;
  end;
  assume
A3: x is Ordinal;
  now
    let y;
A4: y = x implies y c= succ x by XBOOLE_1:7;
A5: now
      assume y in x;
      then
A6:   y c= x by A3,Def2;
      x c= x \/ { x } by XBOOLE_1:7;
      hence y c= succ x by A6;
    end;
    assume y in succ x;
    hence y c= succ x by A1,A5,A4;
  end;
  then
A7: succ x is epsilon-transitive;
  now
    let y,z;
    assume that
A8: y in succ x and
A9: z in succ x;
A10: z in x or z = x by A1,A9;
    y in x or y = x by A1,A8;
    hence y in z or y = z or z in y by A3,A10,Def3;
  end;
  then succ x is epsilon-connected;
  hence thesis by A7;
end;
