Tessellate S.T.E.M.S. (2019) - Computer Science - College - Set 4 - Objective Problem 4

Consider the first-order logic sentence

$\varphi \equiv \exists s \exists t\exists u\forall v\forall w\forall x\forall y \Psi (s,t,u,v,w,x,y)$

where $\Psi (s,t,u,v,w,x,y)$ is a quantifier-free first-order logic formula using only predicate symbols and possibly quality, but no function symbols.

Suppose $\varphi$ has a model with a universe containing 7 elements.

Which one of the following statements is necessarily true?

• A) There exists at least one model of $\varphi$ with universe of size less than or equal to 3.
• B) There exists no model of $\varphi$ with universe of size less than or equal to 3.
• C) There exists no model of $\varphi$ with universe of size greater than 7.
• D) Every model of $\varphi$ has a universe of size equal to 7.

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This problem is a part of Tessellate S.T.E.M.S (2019)