I encountered a formula in a paper, and i need help interpreting it: The original formula as an image, it is in the middle on page 4 of the PDF at the link
i am concerned with the second formula in the image. From earlier in the paper we know that $R_2^{(2)}$ is set union, the $x_1$,$x_2$,... are sets of real numbers, ~ is negation, etc.
Exchanging the symbols for more familiar alternatives i arrive at:
$$\forall A\left( \exists B\neg\left[ A \cup B = B\right]\rightarrow\exists C\exists D\left(\left[\neg C\cup D = D\right]\land \left[A\cup C=A\right]\right)\right)$$
I tried to stay as close to the paper as possible while improving readability. The paper says this statement is false under the given interpretation (sets of real numbers etc), because there are sets with exactly one element. I do not understand this reasoning.
for A = {n} = C, B = D = $\emptyset$, where n is some real number, the formula seems to evaluate to true.
I feel like i might be misinterpreting the scope of the negation operator because it is used once inside, and once outside brackets, in both cases the scope being an equation. assuming common order of precedence shouldn't these be identical? What am i missing?