Does Nash Theorem imply Zermelo's Theorem?

Question

In 2-Player zero-sum game with every information open and no probabilistic strategy required, Nash Theorem states that one of the players has a strategy, in which the player can maintain a situation where he/she can only become worse off. Does this mean that one of the players has a winning(or drawing) strategy? That is, does Nash Theorem imply Zermelo's Theorem?