Not bat an eye / Not bat an eyelid
The idiom is generally used in a negative sense. The word ‘Bat’ came to mean to blink from the early 1900s, possibly originating from the beating of a bird’s wing. Unfortunately, there isn’t a lot of information about the origin of this phrase, but it is a widely used one today.
John can tell a lie without even batting an eyelid.
He ended their relationship without batting an eye. They had been dating for ten years.
My boss did not bat an eyelid when he told me I was fired. I’m so shocked. I thought he liked me!