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

theorem
  x1, x2, x3, x4, x5, x6 are_mutually_distinct & { x1, x2, x3, x4, x5,
  x6 } misses { x7 } implies x1, x2, x3, x4, x5, x6, x7 are_mutually_distinct
proof
  assume that
A1: x1, x2, x3, x4, x5, x6 are_mutually_distinct and
A2: { x1, x2, x3, x4, x5, x6 } misses { x7 };
A3: x1 <> x3 by A1,ZFMISC_1:def 8;
A4: x2 <> x3 by A1,ZFMISC_1:def 8;
A5: x1 <> x6 by A1,ZFMISC_1:def 8;
A6: x1 <> x5 by A1,ZFMISC_1:def 8;
A7: x1 <> x4 by A1,ZFMISC_1:def 8;
A8: not x7 in { x1, x2, x3, x4, x5, x6 } by A2,ZFMISC_1:48;
  then
A9: x7 <> x1 by ENUMSET1:def 4;
A10: x4 <> x6 by A1,ZFMISC_1:def 8;
A11: x4 <> x5 by A1,ZFMISC_1:def 8;
A12: x3 <> x6 by A1,ZFMISC_1:def 8;
A13: x3 <> x5 by A1,ZFMISC_1:def 8;
A14: x3 <> x4 by A1,ZFMISC_1:def 8;
A15: x2 <> x6 by A1,ZFMISC_1:def 8;
A16: x2 <> x5 by A1,ZFMISC_1:def 8;
A17: x2 <> x4 by A1,ZFMISC_1:def 8;
A18: x7 <> x6 by A8,ENUMSET1:def 4;
A19: x5 <> x6 by A1,ZFMISC_1:def 8;
A20: x7 <> x5 by A8,ENUMSET1:def 4;
A21: x7 <> x4 by A8,ENUMSET1:def 4;
A22: x7 <> x3 by A8,ENUMSET1:def 4;
A23: x7 <> x2 by A8,ENUMSET1:def 4;
  x1 <> x2 by A1,ZFMISC_1:def 8;
  hence thesis by A3,A7,A6,A5,A4,A17,A16,A15,A14,A13,A12,A11,A10,A19,A9,A23,A22
,A21,A20,A18,ZFMISC_1:def 9;
end;
