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

theorem
  for A being set, X being Subset of A holds
  {{x} where x is Element of A: x in X} is Subset-Family of A
proof
  let A be set, X be Subset of A;
  {{x} where x is Element of A: x in X} c= bool A
  proof
    let a be object;
     reconsider aa=a as set by TARSKI:1;
    assume a in {{x} where x is Element of A: x in X};
    then ex x being Element of A st ( a = {x})&( x in X);
    then aa c= A by ZFMISC_1:31;
    hence thesis;
  end;
  hence thesis;
end;
