〜なしに・〜なしで
without (asking, warning, a break)
Noun + なしに / なしで / なしでは
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なしに and なしで mean "without": 彼は断りなしに、私の車を使った, "he used my car without asking".
The noun is usually something that should have been there: permission (断り, 許可), a warning (前触れ), a booking (予約), a break (休憩). なしに is more written, and なしで is more conversational.
With a negative, なしでは means "can't do without": スマホなしでは生活できない.
Compare なくして (N1), a formal "without X, Y is impossible", and 抜きで (N2), "leaving something out on purpose". Don't confuse it with なくて, "isn't, and". The tone is often a mild complaint when someone skips a step they should have taken.
Examples
彼は断りなしに、私の車を使った。
He used my car without asking.
この店は予約なしで入れる。
You can get into this restaurant without a booking.
何の前触れもなしに、突然雨が降り出した。
Without any warning, it suddenly started to rain.
休憩なしで、五時間働いた。
I worked five hours without a break.
許可なしに、写真を撮らないでください。
Please don't take photos without permission.