reserve A, B, X, Y, Z, x, y for set;
reserve f for Function;
reserve O for Ordinal;

theorem Th7:
  A is finite & (for X st X in A holds X is finite) iff union A is finite
proof
  thus A is finite & (for X st X in A holds X is finite) implies
  union A is finite by Lm3;
  thus union A is finite implies
  A is finite & for X st X in A holds X is finite
  proof
    assume
A1: union A is finite;
    A c= bool union A by ZFMISC_1:82;
    hence A is finite by A1;
    let X;
    assume X in A;
    then X c= union A by ZFMISC_1:74;
    hence thesis by A1;
  end;
end;
