 reserve X, Y for set, A for Ordinal;

theorem Th8:
  succRel(4) = {[0,1], [1,2], [2,3]}
proof
  now
    let z be object;
    thus z in succRel(4) implies z = [0,1] or z = [1,2] or z = [2,3]
    proof
      assume A1: z in succRel(4);
      then consider x,y being object such that
        A2: z = [x,y] by RELAT_1:def 1;
      reconsider a=x,b=y as set by TARSKI:1;
      [a,b] in succRel(4) by A1, A2;
      then A3: a in 4 & b in 4 & b = succ a by Def1;
      then per cases by CARD_1:52, ENUMSET1:def 2;
      suppose a = 0;
        hence thesis by A2, A3;
      end;
      suppose a = 1;
        hence thesis by A2, A3;
      end;
      suppose a = 2;
        hence thesis by A2, A3;
      end;
      suppose a = 3;
        hence thesis by A3; :: by contradiction;
      end;
    end;
    assume A7: z = [0,1] or z = [1,2] or z = [2,3];
    1 = succ 0 & 1 in 4 & 2 = succ 1 & 2 in 4 & 3 = succ 2 & 3 in 4
      by CARD_1:52, ENUMSET1:def 2;
    hence z in succRel(4) by A7, Th5;
  end;
  hence thesis by ENUMSET1:def 1;
end;
