theorem     :: ZFMISC_1:37
  {x} c= X iff x in X
proof
  thus {x} c= X implies x in X
  proof
    assume
A1: {x} c= X;
    let i;
    assume
A2: i in I;
    then {x}.i c= X.i by A1;
    then {x.i} c= X.i by A2,Def1;
    hence thesis by ZFMISC_1:31;
  end;
  assume
A3: x in X;
  let i;
  assume
A4: i in I;
  then x.i in X.i by A3;
  then {x.i} c= X.i by ZFMISC_1:31;
  hence thesis by A4,Def1;
end;
