Prove undecidable and recognizable

Is there a way that I can use If $L=\big\{\langle M_1,M_2\rangle\mid M_1, M_2\text{ are TM and } L(M_1)\cup L(M_1)=\Sigma^* \big\}$ is in $RE$ or $coRE$ or not in $RE\cup coRE$? to prove that $$L=\big\{\langle M_1,M_2\rangle\mid M_1, M_2\text{ are TM and } L(M_1)\cup L(M_2)\neq\emptyset \big\}$$ is undecidable and recognizable?

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To prove that the language is recognizable simply enumerate all words $$w_1, w_2, \dots,$$ and execute $$L_1$$ and $$L_2$$ on $$w$$ in dovetail fashion (perform one step of $$M_1$$ and $$M_2$$ on $$w_1$$; perform one step of $$M_1$$ and $$M_2$$ on $$w_2$$ and one additional step on $$w_1$$; perform one step of $$M_1$$ and $$M_2$$ on $$w_3$$, one more step on $$w_2$$, and one more step on $$w_1$$; etc). Halt and accept whenever $$M_1$$ or $$M_2$$ accepts.
To prove that the language is not decidable pick $$M_2$$ as the Turing machine that always rejects and notice that $$\langle M_1, M_2 \rangle \in L$$ if and only if $$L(M_1) \neq \emptyset$$. The problem of determining whether a Turing machine accepts at least one word is undecidable (as it can be seen from a simple reduction from the halting problem).