Let $L$ be a first-order language. A set $\Sigma$ of $L$-sentences is witnessing if for every sentence in $\Sigma$ of the form $(\exists v) \xi(v)$, there exists a closed $L$-term $\lambda$ with $\xi(\lambda) \in \Sigma$.
I want to show that any complete consistent witnessing set of sentences has a closed term.
Hint: Consider the sentence $\chi$ that says $(\exists v) (v=v)$.
My attempt: The result is trivial if $\chi \in \Sigma$. I'm not sure what happens if $\chi \notin \Sigma$. We might only have that $\Sigma \vdash \chi$, or even this may not hold. If it does not, then there is a model for $\Sigma$ that does not model $\chi$, and therefore has to be empty.
But where do I go from here?