Is 'Some of x are true' a negation of 'All of x are true'? I don't think this necessarily means there exists a false x, just that at least some x are true.

Is my logic correct here in assuming this is NOT a negation?
The negation of "all $x$ are true" is "some $x$ is false (in the sense that there exists $x$ false, at least one is false)".
$$\left(\bigcap A_i\right)' = \bigcup(A_i') \neq \bigcup A_i$$