
theorem Th4:
  for X being set holds 2 c= card X iff X is non trivial
proof
  let X be set;
  set z = the Element of X;
  thus 2 c= card X implies X is non trivial
  proof
    assume 2 c= card X;
    then consider x,y being object such that
A1: x in X and
A2: y in X and
A3: x<>y by Th2;
    now
      given z being object such that
A4:   X={z};
      thus x = z by A1,A4,TARSKI:def 1
        .= y by A2,A4,TARSKI:def 1;
    end;
    hence thesis by A1,A3,ZFMISC_1:131;
  end;
  assume
A5: X is non trivial;
  then X c= {z} implies X={z};
  then consider w being object such that
A6: w in X and
A7: not w in {z} by A5;
  w <> z by A7,TARSKI:def 1;
  hence thesis by A6,Th2;
end;
