reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;

theorem Th50:
  for X being non empty set st (for x st x in X holds x is epsilon Ordinal) &
    for a st a in X holds first_epsilon_greater_than a in X
  holds union X is epsilon Ordinal
  proof
    let X be non empty set such that
A1: for x st x in X holds x is epsilon Ordinal and
A2: for a st a in X holds first_epsilon_greater_than a in X;
    now
      let x; assume
A3:   x in X;
      hence x is epsilon Ordinal by A1;
      reconsider a = x as Ordinal by A1,A3;
      take e = first_epsilon_greater_than a;
      thus x in e & e in X by A2,A3,Def6;
    end;
    hence union X is epsilon Ordinal by Th49;
  end;
