Unless I'm mistaken, the translation in "Matrix" was "Łyżka nie istnieje." (The spoon doesn't exist)
It sure sounds better as a standalone phrase.
Maybe "Nie ma żadnej łyżki" (more like 'There IS NO spoon') would sound okayish.
Why isn't it moją łyżką? I thought nouns took the instrumental case after jest?
Nouns in Instrumental occur when there are at least two nouns. One of them should appear in Nominative and another one in Instrumental and often the word functions are not interchangeable. The noun in Instrumental can be treated as a complement of the verb "być": https://en.wikipedia.org/wiki/Copula_%28linguistics%29#Slavic
Here you can see so-called existential "być" which corresponds to English "there is/are" and it refers to the existence or presence of something: https://en.wikipedia.org/wiki/Existential_clause