Does repeated vertex in polygon definition makes it self-intersecting?

Question

If we consider polygon definition as list of connected vertices (where we don't have sequences of same repeated points) and we have equal vertices (with same coordinates) in different position $$ P = (v_0, v_1, \dots, v_M) \quad \exists i, j \in \{0, \dots, M\}: P_i = P_j $$ -- is it enough to conclude that polygon is self-intersecting? I cannot think of counterexample and it seems kind of obvious, so I have thought there should be some related lemma/theorem or a logical proof.