
theorem Th5:
  for X being set holds 3 c= card X iff ex x,y,z being object st x in
  X & y in X & z in X & x<>y & x<>z & y<>z
proof
  let X be set;
  thus 3 c= card X implies
    ex x,y,z being object st x in X & y in X & z in X & x
  <>y & x<>z & y<>z
  proof
    assume 3 c= card X;
    then card 3 c= card X;
    then consider f being Function such that
A1: f is one-to-one and
A2: dom f = 3 and
A3: rng f c= X by CARD_1:10;
    take x=f.0;
    take y=f.1;
    take z=f.2;
A4: 0 in dom f by A2,ENUMSET1:def 1,YELLOW11:1;
    then f.0 in rng f by FUNCT_1:def 3;
    hence x in X by A3;
A5: 1 in dom f by A2,ENUMSET1:def 1,YELLOW11:1;
    then f.1 in rng f by FUNCT_1:def 3;
    hence y in X by A3;
A6: 2 in dom f by A2,ENUMSET1:def 1,YELLOW11:1;
    then f.2 in rng f by FUNCT_1:def 3;
    hence z in X by A3;
    thus x <> y by A1,A4,A5,FUNCT_1:def 4;
    thus x <> z by A1,A4,A6,FUNCT_1:def 4;
    thus thesis by A1,A5,A6,FUNCT_1:def 4;
  end;
  given x,y,z being object such that
A7: x in X & y in X & z in X and
A8: x<>y & x<>z & y<>z;
  {x,y,z} c= X
  by A7,ENUMSET1:def 1;
  then card {x,y,z} c= card X by CARD_1:11;
  hence thesis by A8,CARD_2:58;
end;
