What does "notwithstanding" mean in the Supremacy Clause of the U.S. Constitution?

I suggest that in the Supremacy Clause of the US Constitution the word "notwithstanding" at the end of the sentence concisely and essentially means "does not exist".

This Constitution, and the Laws of the United States […] shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, any Thing in the Constitution or Laws of any State to the Contrary notwithstanding.

This should be understood in the light of The Constitution and federal law being the final word on any constitutional or legal subject.

The preemption doctrine originates from the supremacy clause of Article 6 of the U.S. Constitution. This doctrine states that any federal law, even if it is only a regulation from a federal agency, supersedes any conflicting state law, even if that law is part of the state's constitution. See https://smallbusiness.chron.com/state-labor-laws-supercede-federal-labor-laws-56969.html

Thus if the Constitution or federal law says you must eat cabbage once a day, but Arkansas State law or constitution says cabbage is illegal, then the Arkansas law is invalid.

Thus

This Constitution, and Federal Law […] shall be the supreme Law of the Land; and the Judges in every State shall be bound thereby, even if there is a local, state, or city law that claims to be superior to, or dismiss, or be contrary to the Constitution, and the Laws of the United States.