reserve x,y for object,X,Y,A,B,C,M for set;
reserve P,Q,R,R1,R2 for Relation;

theorem Th2:
  X = {} iff {_{X}_} = {}
proof
  thus X = {} implies {_{X}_} = {}
  proof
    assume
A1: X = {};
    assume not thesis;
    then consider y being object such that
A2: y in {_{X}_} by XBOOLE_0:def 1;
    ex x being object st ( y = {x})&( x in X) by A2,Th1;
    hence contradiction by A1;
  end;
  assume
A3: {_{X}_} = {};
  assume not thesis;
  then consider x being object such that
A4: x in X by XBOOLE_0:def 1;
  {x} in {_{X}_} by A4,Th1;
  hence contradiction by A3;
end;
