reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;
reserve f for Ordinal-Sequence;
reserve U,W for Universe;

theorem Th41:
  a in U implies for f being Ordinal-Sequence of a,U holds Union f in U
  proof assume
A1: a in U;
    let f be Ordinal-Sequence of a,U;
    dom f = a by FUNCT_2:def 1; then
    card dom f in card U & card rng f c= card dom f & rng f c= On U & On U c= U
    by A1,CARD_2:61,CLASSES2:1,ORDINAL2:7,RELAT_1:def 19; then
    card rng f in card U & rng f c= U by ORDINAL1:12; then
    rng f in U by CLASSES1:1;
    hence Union f in U by CLASSES2:59;
  end;
