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

theorem Th1:
  y in {_{X}_} iff ex x st y = {x} & x in X
proof
  thus y in {_{X}_} implies ex x st y = {x} & x in X
  proof
    assume
A1: y in {_{X}_};
    per cases;
    suppose
A2:   X is empty;
      ex x being object st x in X & y = Class(id X,x) by A1,EQREL_1:def 3;
      hence thesis by A2;
    end;
    suppose X is non empty;
      then reconsider X9 = X as non empty set;
      y in the set of all {x} where x is Element of X9 by A1,EQREL_1:37;
      then ex x being Element of X9 st ( y = {x});
      hence thesis;
    end;
  end;
  given x such that
A3: y = {x} and
A4: x in X;
  reconsider X9 = X as non empty set by A4;
  y in the set of all {z} where z is Element of X9 by A3,A4;
  hence thesis by EQREL_1:37;
end;
