True, correct. The universal set U is characterized by ∀:∈ ∀ x : x ∈ U . Taking the complement yields a set U c that is characterized by ∀:∉ ∀ x : x ∉ U c . This is equivalent to the statement ¬∃:∈ ¬ ∃ x : x ∈ U c and hence U c is an (the) empty set. (Depending on your theory, there may not be a unique empty set.)
