reserve A,A1,A2,B,C,D for Ordinal,
  X,Y for set,
  x,y,a,b,c for object,
  L,L1,L2,L3 for Sequence,
  f for Function;

theorem Th20:
  (for A st A in X holds A in D) implies sup X c= D
proof
  assume
A1: for A st A in X holds A in D;
  On X c= D
  proof
    let x be object;
    assume
A2: x in On X;
    then reconsider A = x as Ordinal by ORDINAL1:def 9;
    A in X by A2,ORDINAL1:def 9;
    hence thesis by A1;
  end;
  hence thesis by Def3;
end;
