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 Th49:
  for X being non empty set st for x st x in X holds x is epsilon Ordinal &
    ex e being epsilon Ordinal st x in e & e in X
  holds union X is epsilon Ordinal
  proof
    let X be non empty set; assume
A1: for x st x in X holds x is epsilon Ordinal &
    ex e being epsilon Ordinal st x in e & e in X; then
    for x st x in X holds x is Ordinal; then
    reconsider a = union X as epsilon-transitive epsilon-connected set
by ORDINAL1:23;
    now
      let b; assume b in a; then
      consider x being set such that
A2:   b in x & x in X by TARSKI:def 4;
      reconsider x as epsilon Ordinal by A2,A1;
      succ b in x by A2,ORDINAL1:28;
      hence succ b in a by A2,TARSKI:def 4;
    end; then
A3: a is limit_ordinal by ORDINAL1:28;
    set z = the Element of X;
    ex e being epsilon Ordinal st z in e & e in X by A1; then
A4: a <> {} by TARSKI:def 4;
    a is epsilon
    proof
      thus exp(omega,a) c= a
      proof
        let x be Ordinal; assume x in exp(omega,a); then
        consider b such that
A5:     b in a & x in exp(omega,b) by A3,A4,Th11;
        consider y being set such that
A6:     b in y & y in X by A5,TARSKI:def 4;
        reconsider y as epsilon Ordinal by A1,A6;
        exp(omega,b) in exp(omega,y) by A6,ORDINAL4:24; then
        exp(omega,b) in y by Def5; then
        x in y by A5,ORDINAL1:10;
        hence thesis by A6,TARSKI:def 4;
      end;
      thus thesis by ORDINAL4:32;
    end;
    hence thesis;
  end;
