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

theorem Th2:
  for x1, x2, x3, x4, x5, x6 being set st x1, x2, x3, x4, x5, x6
  are_mutually_distinct holds card {x1, x2, x3, x4, x5, x6} = 6
proof
  let x1, x2, x3, x4, x5, x6 be set;
A1: {x1,x2,x3,x4,x5,x6} = {x1,x2,x3,x4,x5} \/ {x6} by ENUMSET1:15;
  assume
A2: x1, x2, x3, x4, x5, x6 are_mutually_distinct;
  then
A3: x1 <> x3 by ZFMISC_1:def 8;
A4: x4 <> x5 by A2,ZFMISC_1:def 8;
A5: x3 <> x5 by A2,ZFMISC_1:def 8;
A6: x3 <> x4 by A2,ZFMISC_1:def 8;
A7: x2 <> x5 by A2,ZFMISC_1:def 8;
A8: x2 <> x4 by A2,ZFMISC_1:def 8;
A9: x2 <> x3 by A2,ZFMISC_1:def 8;
A10: x1 <> x5 by A2,ZFMISC_1:def 8;
A11: x1 <> x4 by A2,ZFMISC_1:def 8;
  x1 <> x2 by A2,ZFMISC_1:def 8;
  then x1, x2, x3, x4, x5 are_mutually_distinct by A3,A11,A10,A9,A8,A7,A6,A5
,A4,ZFMISC_1:def 7;
  then
A12: card {x1,x2,x3,x4,x5} = 5 by CARD_2:63;
A13: x3 <> x6 by A2,ZFMISC_1:def 8;
A14: x2 <> x6 by A2,ZFMISC_1:def 8;
A15: x5 <> x6 by A2,ZFMISC_1:def 8;
A16: x4 <> x6 by A2,ZFMISC_1:def 8;
  x1 <> x6 by A2,ZFMISC_1:def 8;
  then not x6 in {x1,x2,x3,x4,x5} by A14,A13,A16,A15,ENUMSET1:def 3;
  hence card {x1,x2,x3,x4,x5,x6} = 5+1 by A12,A1,CARD_2:41
    .= 6;
end;
