reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem Th4:
  { x1, x2, x3 } misses { x4, x5, x6 } implies x1 <> x4 & x1 <> x5
  & x1 <> x6 & x2 <> x4 & x2 <> x5 & x2 <> x6 & x3 <> x4 & x3 <> x5 & x3 <> x6
proof
  assume
A1: { x1, x2, x3 } misses { x4, x5, x6 };
A2: x5 in { x4, x5, x6 } by ENUMSET1:def 1;
  assume x1 = x4 or x1 = x5 or x1 = x6 or x2 = x4 or x2 = x5 or x2 = x6 or x3
  = x4 or x3 = x5 or x3 = x6;
  then
A3: x4 in { x1, x2, x3 } or x5 in { x1, x2, x3 } or x6 in { x1, x2, x3 } by
ENUMSET1:def 1;
A4: x6 in { x4, x5, x6 } by ENUMSET1:def 1;
  x4 in { x4, x5, x6 } by ENUMSET1:def 1;
  hence thesis by A1,A3,A2,A4,XBOOLE_0:3;
end;
