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Adjective: bigheaded  ,big'he-did
  1. (used colloquially) overly conceited or arrogant
    "they're snobs--stuck-up and uppity and bigheaded";
    - persnickety, snooty [informal], snot-nosed [informal], snotty [informal], stuck-up [informal], too big for one's breeches [informal], uppish [informal], snobbish

See also: proud, toffee-nosed [Brit, informal]