
theorem Th10:
  for A being set, e1,e2 being set st {e1,e2} in TWOELEMENTSETS(A)
  holds e1 in A & e2 in A & e1<>e2
proof
  let A be set, e1,e2 be set;
  assume
A1: {e1,e2} in TWOELEMENTSETS(A);
  then consider x,y being object such that
A2: x in A & y in A & not x=y and
A3: {e1,e2} = {x,y} by Th8;
  per cases by A3,ZFMISC_1:6;
  suppose
    e1=x & e2=x;
    then {x} in TWOELEMENTSETS(A) by A1,ENUMSET1:29;
    then ex x1,x2 being object
st x1 in A & x2 in A &( not x1=x2)& { x} = {x1,x2} by Th8;
    hence thesis by ZFMISC_1:5;
  end;
  suppose
    e1=x & e2=y;
    hence thesis by A2;
  end;
  suppose
    e1=y & e2=x;
    hence thesis by A2;
  end;
  suppose
    e1=y & e2=y;
    then {y} in TWOELEMENTSETS(A) by A1,ENUMSET1:29;
    then
    ex x1,x2 being object st x1 in A & x2 in A &( not x1=x2)& { y}={x1,x2}
by Th8;
    hence thesis by ZFMISC_1:5;
  end;
end;
