
theorem Th20:
  for X being set holds 4 c= card X iff
    ex w,x,y,z being object st w in X & x in X & y in X & z in X &
      w<>x & w<>y & w<>z & x<>y & x<>z & y<>z
proof
  let X be set;
  thus 4 c= card X implies
    ex w,x,y,z being object st w in X & x in X & y in X & z in X &
      w<>x & w<>y & w <> z & x<>y & x<>z & y<>z
  proof
    assume 4 c= card X;
    then card 4 c= card X;
    then consider f being Function such that
A1: f is one-to-one and
A2: dom f = 4 and
A3: rng f c= X by CARD_1:10;
    take w=f.0;
    take x=f.1;
    take y=f.2;
    take z=f.3;
A4: 0 in dom f by A2,ENUMSET1:def 2,CARD_1:52;
    then f.0 in rng f by FUNCT_1:def 3;
    hence w in X by A3;
A5: 1 in dom f by A2,ENUMSET1:def 2,CARD_1:52;
    then f.1 in rng f by FUNCT_1:def 3;
    hence x in X by A3;
A6: 2 in dom f by A2,ENUMSET1:def 2,CARD_1:52;
    then f.2 in rng f by FUNCT_1:def 3;
    hence y in X by A3;
A7: 3 in dom f by A2,ENUMSET1:def 2,CARD_1:52;
    then f.3 in rng f by FUNCT_1:def 3;
    hence z in X by A3;
    thus w <> x by A1,A4,A5,FUNCT_1:def 4;
    thus w <> y by A1,A4,A6,FUNCT_1:def 4;
    thus w <> z by A1,A4,A7,FUNCT_1:def 4;
    thus x <> y by A1,A5,A6,FUNCT_1:def 4;
    thus x <> z by A1,A5,A7,FUNCT_1:def 4;
    thus thesis by A1,A6,A7,FUNCT_1:def 4;
  end;
  given w,x,y,z being object such that
A8: w in X & x in X & y in X & z in X and
A9: w<>x & w<>y & w<>z & x<>y & x<>z & y<>z;
  for a being object st a in {w,x,y,z} holds a in X by A8,ENUMSET1:def 2;
  then card {w,x,y,z} c= card X by TARSKI:def 3, CARD_1:11;
  hence thesis by A9, CARD_2:59;
end;
