Consider the following property of recursively enumberable (RE) languages $$ L = \{ J \in \text{RE} \mid J \cap L_{uni} \ne \emptyset \}$$where $L_{uni}$ is the language of the Universal Turing machine.I am wondering if $L$ is semi-decidable. That is (by definition) if the set $$ S = \{ \langle T \rangle \mid L(T) \in L\}$$ is recursively enumerable.If I am not mistaken then this should be true because one can always construct a Turing machine $M$ that accepts a code $\langle T \rangle$ and  (using non-determinism) generates the right input for $T$ and $T_{uni}$, verifies that both accept the generated word (simulating the machines) and if so accepts the supplied string.Can someone verify this claim?
Question about a semi-decidable property of languages
computability;turing machines