
theorem
  for A being 1-element set, B being set st A /\ B is non empty
  holds A c= B
proof
  let A be 1-element set, B be set;
A1: A /\ B c= B by XBOOLE_1:17;
  assume A /\ B is non empty;
  then consider a being object such that
A2: a in A /\ B;
A3: ex s being Element of A st A = {s} by SUBSET_1:46;
  A /\ B c= A by XBOOLE_1:17;
  then {a} c= A by A2,ZFMISC_1:31;
  then {a} = A by A3,ZFMISC_1:18;
  hence thesis by A2,A1,ZFMISC_1:31;
end;
