 reserve X, Y for set, A for Ordinal;

theorem Th6:
  succRel(2) = {[0,1]}
proof
  now
    let z be object;
    hereby
      assume A1: z in succRel(2);
      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(2) by A1, A2;
      then A3: a in 2 & b in 2 & b = succ a by Def1;
      then per cases by CARD_1:50, TARSKI:def 2;
      suppose a = 0;
        hence z = [0,1] by A2, A3;
      end;
      suppose a = 1;
        hence z = [0,1] by A3; :: by contradiction;
      end;
    end;
    assume A5: z = [0,1];
    1 = succ 0 & 1 in 2 by CARD_1:50, TARSKI:def 2;
    hence z in succRel(2) by A5, Th4;
  end;
  hence thesis by TARSKI:def 1;
end;
