A slow start to the contest for the hosts allowed Mexico to carve out the first real opportunity after 25 minutes, with Hector Herrera beating two defenders before getting a shot away at goal which was kept out by Julio Cesar.
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Two minutes later Brazil came alive as Neymar planted a header towards the bottom corner of the net, only for Ochoa to pull off the first of many top-quality stops to deny the forward.
A cross into the box moments before the end of the half almost provided the breakthrough goal when David Luiz found himself unmarked inside the box, but Ochoa was once more on hand to pull off a close-range save to deny Brazil.
Bernard was brought on at the interval in place of Ramires to add some extra attacking intent for Luiz Felipe Scolari's side, and he very nearly made an instant impact when he delivered a cross for Neymar, which was well cleared by Francisco Rodriguez for a corner.
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Neymar, who bagged two goals in his side's opening game against Croatia last week, collected a cross into the box from Bernard on the 70-minute mark before seeing his resulting strike kept out yet again by Ochoa.
Bernard was provider once more shortly after when he picked out fellow substitute Jo with a low pass, but his shot failed to test the Mexican stopper.
As the game edged towards full time, Ochoa was called into action for a final time to deny Silva from a close-range header to ensure that the game ended goalless.
Both sides move up to four points in Group A following the draw, meaning that either Croatia or Cameroon will exit the competition if they lose tomorrow afternoon.