Why do people think expecting men to hold doors for women and to let them go first isn’t sexist just because it’s good manners?

Yes... Good manners and basic kindness and consideration applies to everyone regardless of sex (or anything else for that matter)... If you are the 1st person to the door... open it and hold it open for whoever is behind you/with you... But equality doesn't mean that chivalry is dead & being chivalrous doesn't mean you are being sexist. 

Expecting men to act like gentlemen while not expecting women to act like ladies, is sexist. It is pushing a social responsibility onto men only while not making any similar expectation of women. That is sexism.

And it is a sexist society that shames men into this expectation that their traditional responsibilities are expected of them while allowing feminists to frame any expectation of women into their previous responsibilities as being "sexist". This is the currently sexist state of Western society.

Men are supposed to "man up" to their traditional responsibilities while women get the "choice" of whether they would like to participate in their traditional responsibilities or not. Of course this is sexist!

As an aside, a response to the person who states that you keep posting this, is that it keeps being reposted because it keeps being falsely reported and deleted, for no good reason.

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You keep on posting this.  You really need to get a life!

Holding a door open for the person behind you (not necessarily letting them go first) is just basic good manners.  It doesn't matter if they - or you - are male or female.