reserve W for Universe,
  H for ZF-formula,
  x,y,z,X for set,
  k for Variable,
  f for Function of VAR,W,
  u,v for Element of W;
reserve F for Function,
  A,B,C for Ordinal,
  a,b,b1,b2,c for Ordinal of W,
  fi for Ordinal-Sequence,
  phi for Ordinal-Sequence of W,
  H for ZF-formula;
reserve psi for Ordinal-Sequence;
reserve L for DOMAIN-Sequence of W,
  n for Element of NAT,
  f for Function of VAR,L.a;

theorem Th19:
  X in W implies sup X in W
proof
  reconsider a = union On X as Ordinal by ORDINAL3:5;
A1: On X c= X by ORDINAL2:7;
  assume X in W;
  then On X in W by A1,CLASSES1:def 1;
  then reconsider a as Ordinal of W by CLASSES2:59;
  sup X c= succ a by ORDINAL3:72;
  hence thesis by CLASSES1:def 1;
end;
