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

theorem Th6:
  X c= Y iff {_{X}_} c= {_{Y}_}
proof
  thus X c= Y implies {_{X}_} c= {_{Y}_}
  proof
    assume
A1: X c= Y;
    let y be object;
    assume y in {_{X}_};
    then ex x being object st ( y = {x})&( x in X) by Th1;
    hence thesis by A1,Th1;
  end;
  assume
A2: {_{X}_} c= {_{Y}_};
  let y be object;
  assume y in X;
  then {y} in {_{X}_} by Th1;
  then ex x1 being object st ( {y} = {x1})&( x1 in Y) by A2,Th1;
  hence thesis by ZFMISC_1:3;
end;
