i think it's the opposite. in FPTP system the largest minority (of voters) wins. if you vote against one candidate, it will (probably) create/be another minority. to make sure the candidate loses, the largest minority have to agree for another candidate, just voting any other candidate won't do. related cgp grey's video - https://yewtu.be/watch?v=s7tWHJfhiyo.
edited to clarify.
lets assume the election results as:
candidate A - 20%
candidate B - 35%
candidate C - 15%
candidate D - 25%
candidate E - 05%
candidate B won with only 35% voting for it while 65% voted against candidate B. clealy the majority of people voted against candidate B, but that doesn't matter as in FPTP, not majority but largest minority (35% that voted for candidate B) wins.
thus, i think you vote for not against in FPTP voting system.