不共戴天

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

CEDICT

to be absolutely irreconcilable

Wiktionary

不願與仇人共生世間。語本《禮記．曲禮上》：「父之讎，弗與共戴天。」比喻仇恨極深。《三國演義．第三九回》：「凌統叩頭大哭曰：『不共戴天之讎豈容不報！』」《儒林外史．第四一回》：「我父親和他涉了訟，他買囑知縣，將我父親斷輸了，這是我不共戴天之仇。」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