reserve X for set;

theorem
  for X being non empty set holds InclPoset X is upper-bounded implies
  union X in X
proof
  let X be non empty set;
  assume InclPoset X is upper-bounded;
  then consider x be Element of InclPoset X such that
A1: x is_>=_than the carrier of InclPoset X by YELLOW_0:def 5;
  now
    let y be object;
    assume y in union X;
    then consider Y be set such that
A2: y in Y and
A3: Y in X by TARSKI:def 4;
    reconsider Y as Element of InclPoset X by A3;
    Y <= x by A1;
    then Y c= x by Th3;
    hence y in x by A2;
  end;
  then
A4: union X c= x;
  x in X & x c= union X by ZFMISC_1:74;
  hence thesis by A4,XBOOLE_0:def 10;
end;
