
theorem Th11:
  for X being set st X is mutually-disjoint
  holds X \ {{}} is a_partition of union X
proof
  let X be set;
  assume A1: X is mutually-disjoint;
  now
    let x be object;
    reconsider y = x as set by TARSKI:1;
    assume x in X \ {{}};
    then y c= union X by ZFMISC_1:74;
    hence x in bool union X;
  end;
  then A2: X \ {{}} is Subset of bool union X by TARSKI:def 3;
  now
    thus union (X \ {{}}) = union X by PARTIT1:2;
    let A be Subset of union X;
    assume A3: A in X \ {{}};
    then not A in {{}} by XBOOLE_0:def 5;
    hence A <> {} by TARSKI:def 1;
    let B be Subset of union X;
    assume B in X \ {{}} & A <> B;
    hence A misses B by A1, A3, TAXONOM2:def 5;
  end;
  hence thesis by A2, EQREL_1:def 4;
end;
