不共戴天

bùgòngdàitiān
(idiom) (of enemies) cannot live under the same sky; absolutely irreconcilable

CEDICT

to be absolutely irreconcilable

Wiktionary

不愿与仇人共生世间。语本《礼记．曲礼上》：「父之雠，弗与共戴天。」比喻仇恨极深。《三国演义．第三九回》：「凌统叩头大哭曰：『不共戴天之雠岂容不报！』」《儒林外史．第四一回》：「我父亲和他涉了讼，他买嘱知县，将我父亲断输了，这是我不共戴天之仇。」也作「不同日月」、「不共天地」。

重編國語辭典

Entry Composition

1 不 bù
  1. no; not so
  2. (bound form) not; un-
2 共戴 Gòngdài the Gongdai era (a Chinese era name; the era of Bogd Khan as khagan of the Bogd Khanate of Mongolia, from 1911 to 1915, and then as restored from 1921 to 1924)
3 天 tiān
  1. day
  2. sky
  3. heaven