reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;
reserve x1,x2,x3,x4,x5,x6,x7,x8 for object;

theorem
  x1,x2,x3,x4,x5 are_mutually_distinct implies card {x1,x2,x3,x4,x5} = 5
proof
A1: {x1,x2,x3,x4,x5} = {x1,x2,x3,x4} \/ {x5} by ENUMSET1:10;
  assume
A2: x1,x2,x3,x4,x5 are_mutually_distinct;
  then
A3: x3 <> x5 & x4 <> x5 by ZFMISC_1:def 7;
A4: x2 <> x4 & x3 <> x4 by A2,ZFMISC_1:def 7;
A5: x1 <> x4 & x2 <> x3 by A2,ZFMISC_1:def 7;
  x1 <> x5 & x2 <> x5 by A2,ZFMISC_1:def 7;
  then
A6: not x5 in {x1,x2,x3,x4} by A3,ENUMSET1:def 2;
  x1 <> x2 & x1 <> x3 by A2,ZFMISC_1:def 7;
  then card {x1,x2,x3,x4} = 4 by A5,A4,Th58;
  hence card {x1,x2,x3,x4,x5} = 4+1 by A6,A1,Th40
    .= 5;
end;
