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

theorem
  for x1, x2, x3, x4, x5, x6, x7 being set st x1, x2, x3, x4, x5, x6, x7
  are_mutually_distinct holds card {x1, x2, x3, x4, x5, x6, x7} = 7
proof
  let x1, x2, x3, x4, x5, x6, x7 be set;
A1: {x1,x2,x3,x4,x5,x6,x7} = {x1,x2,x3,x4,x5,x6} \/ {x7} by ENUMSET1:21;
  assume
A2: x1, x2, x3, x4, x5, x6, x7 are_mutually_distinct;
  then
A3: x1 <> x3 by ZFMISC_1:def 9;
A4: x5 <> x7 by A2,ZFMISC_1:def 9;
A5: x4 <> x7 by A2,ZFMISC_1:def 9;
A6: x3 <> x7 by A2,ZFMISC_1:def 9;
A7: x2 <> x7 by A2,ZFMISC_1:def 9;
A8: x4 <> x6 by A2,ZFMISC_1:def 9;
A9: x4 <> x5 by A2,ZFMISC_1:def 9;
A10: x5 <> x6 by A2,ZFMISC_1:def 9;
A11: x1 <> x5 by A2,ZFMISC_1:def 9;
A12: x1 <> x4 by A2,ZFMISC_1:def 9;
A13: x3 <> x6 by A2,ZFMISC_1:def 9;
A14: x3 <> x5 by A2,ZFMISC_1:def 9;
A15: x3 <> x4 by A2,ZFMISC_1:def 9;
A16: x2 <> x6 by A2,ZFMISC_1:def 9;
A17: x2 <> x5 by A2,ZFMISC_1:def 9;
A18: x2 <> x4 by A2,ZFMISC_1:def 9;
A19: x2 <> x3 by A2,ZFMISC_1:def 9;
A20: x1 <> x6 by A2,ZFMISC_1:def 9;
  x1 <> x2 by A2,ZFMISC_1:def 9;
  then x1, x2, x3, x4, x5, x6 are_mutually_distinct by A3,A12,A11,A20,A19,A18
,A17,A16,A15,A14,A13,A9,A8,A10,ZFMISC_1:def 8;
  then
A21: card {x1,x2,x3,x4,x5,x6} = 6 by Th2;
A22: x6 <> x7 by A2,ZFMISC_1:def 9;
  x1 <> x7 by A2,ZFMISC_1:def 9;
  then not x7 in {x1,x2,x3,x4,x5,x6} by A7,A6,A5,A4,A22,ENUMSET1:def 4;
  hence card {x1,x2,x3,x4,x5,x6,x7} = 6+1 by A21,A1,CARD_2:41
    .= 7;
end;
