reserve a,b,c,d,x,y,w,z,x1,x2,x3,x4 , X for set;
reserve A for non empty set;

theorem
  {x1,x2,x3,x4} c= X iff x1 in X & x2 in X & x3 in X & x4 in X
proof
A1: x1 in {x1,x2,x3,x4} by ENUMSET1:def 2;
A2: x2 in {x1,x2,x3,x4} by ENUMSET1:def 2;
A3: x3 in {x1,x2,x3,x4} by ENUMSET1:def 2;
  x4 in {x1,x2,x3,x4} by ENUMSET1:def 2;
  hence {x1,x2,x3,x4} c= X implies x1 in X & x2 in X & x3 in X & x4 in X
    by A1,A2,A3;
  assume that
A4: x1 in X and
A5: x2 in X and
A6: x3 in X and
A7: x4 in X;
  let z be object;
  assume z in {x1,x2,x3,x4};
  hence thesis by A4,A5,A6,A7,ENUMSET1:def 2;
end;
