 reserve X, Y for set, A for Ordinal;

theorem Th7:
  succRel(3) = {[0,1], [1,2]}
proof
  now
    let z be object;
    thus z in succRel(3) implies z = [0,1] or z = [1,2]
    proof
      assume A1: z in succRel(3);
      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(3) by A1, A2;
      then A3: a in 3 & b in 3 & b = succ a by Def1;
      then per cases by CARD_1:51, ENUMSET1:def 1;
      suppose a = 0;
        hence thesis by A2, A3;
      end;
      suppose a = 1;
        hence thesis by A2, A3;
      end;
      suppose a = 2;
        hence thesis by A3; :: by contradiction;
      end;
    end;
    assume A6: z = [0,1] or z = [1,2];
    1 = succ 0 & 1 in 3 & 2 = succ 1 & 2 in 3 by CARD_1:51, ENUMSET1:def 1;
    hence z in succRel(3) by A6, Th5;
  end;
  hence thesis by TARSKI:def 2;
end;
