
theorem Th6:
for x being set holds singletons {x} = { {x} }
proof
 let x be set;
A1: {x} c= {x};
 thus singletons {x} c= { {x} } proof
   let a be object;
   assume a in singletons {x};
    then consider f being Subset of {x} such that
   A2: a = f and
   A3: f is 1-element;
    f = {} or f = {x} by ZFMISC_1:33;
   hence thesis by A2,A3,TARSKI:def 1;
 end;
   let a be object;
   assume a in {{x}};
   then a = {x} by TARSKI:def 1;
   hence thesis by A1;
end;
