reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;

theorem
  X in Rank A implies union X in Rank A
proof
  assume
A1: X in Rank A;
A2: now
    given B such that
A3: A = succ B;
A4: union X c= union Rank B by ZFMISC_1:77,A1,A3,Th32;
 union Rank B c= Rank B by Th34;
    hence thesis by A3,Th32,A4,XBOOLE_1:1;
  end;
 now
    assume that
A5: A <> {} and
A6: for B holds A <> succ B;
A7: A is limit_ordinal by A6,ORDINAL1:29;
    then consider B such that
A8: B in A and
A9: X in Rank B by A1,A5,Th31;
A10: union X c= union Rank B by ZFMISC_1:77,A9,ORDINAL1:def 2;
A11: union Rank B c= Rank B by Th34;
     succ B <> A by A6;
then A12: succ B c< A by A8,ORDINAL1:21;
A13: union X in Rank succ B by A11,Th32,A10,XBOOLE_1:1;
    succ B in A by A12,ORDINAL1:11;
    hence thesis by A7,A13,Th31;
  end;
  hence thesis by A1,A2,Lm2;
end;
