
theorem Th25:
for x being set holds { {}, {x} } is SimpleGraph
proof
  let x be set;
   set H = { {}, {x} };
A1: H is 1-at_most_dimensional
proof
    let a be set such that
    A2: a in H;
    per cases by A2,TARSKI:def 2;
    suppose a = {};
      hence card a c= 1+1;
    end;
    suppose a = {x};
      then
A3:      card a = 1 by CARD_1:30;
        Segm 1 c= Segm(1+1) by NAT_1:39;
      hence card a c= 1+1 by A3;
    end;
   end;
   H is subset-closed proof
    let X,Y be set such that
   A4: X in H and
   A5: Y c= X;
    per cases by A4,TARSKI:def 2;
    suppose X = {};
      then Y = {} by A5;
      hence Y in H by TARSKI:def 2;
    end;
    suppose A6: X = {x};
       per cases by A6,A5,ZFMISC_1:33;
       suppose Y = {};
         hence Y in H by TARSKI:def 2;
       end;
       suppose Y = {x};
         hence Y in H by TARSKI:def 2;
       end;
    end;
   end;
  hence { {}, {x} } is SimpleGraph by A1;
end;
