reserve N for Cardinal;
reserve M for Aleph;
reserve X for non empty set;
reserve Y,Z,Z1,Z2,Y1,Y2,Y3,Y4 for Subset of X;
reserve S for Subset-Family of X;
reserve x for set;
reserve F,Uf for Filter of X;
reserve S for non empty Subset-Family of X;
reserve I for Ideal of X;
reserve S,S1 for Subset-Family of X;
reserve FS for non empty Subset of Filters(X);
reserve X for infinite set;
reserve Y,Y1,Y2,Z for Subset of X;
reserve F,Uf for Filter of X;
reserve x for Element of X;

theorem Th32:
  for A being limit_ordinal Ordinal for X being set st X c= A
  holds sup X = A implies union X = A
proof
  let A be limit_ordinal Ordinal;
  let X be set;
  assume X c= A;
  then
A1: union X c= union A by ZFMISC_1:77;
  assume
A2: sup X = A;
  thus union X c= A by A1,ORDINAL1:def 6;
  thus A c= union X
  proof
    let X1 be object such that
A3: X1 in A;
    reconsider X2=X1 as Ordinal by A3;
    succ X2 in A by A3,ORDINAL1:28;
    then
A4: ex B being Ordinal st B in X & succ X2 c= B by A2,ORDINAL2:21;
    X2 in succ X2 by ORDINAL1:6;
    hence thesis by A4,TARSKI:def 4;
  end;
end;
