Constructing a reduction between two languages about pairs of Turing machines

I'm curious about a potential relation between the following two languages.

$$L_1 := \{\langle M_1, M_2 \rangle : L(M_1) \cap L(M_2) \ne \emptyset \}$$.

$$L_2 := \{\langle M_1, M_2 \rangle : L(M_1) \ne L(M_2) \}$$.

Is it true that $$L_2$$ mapping reduces to $$L_1$$?

I've had many failed attempts to construct a reduction and would appreciate any help.

• Show that $$L_2$$ is not computably enumerable.
• Show that $$L_1$$ is computably enumerable.
• If $$L_2$$ mapping reduces to $$L_1$$, what does it imply?
• Are you sure that $L_2$ is not computably enumerable? – CuriousKid7 Mar 20 at 0:24
• Try enumerating the pairs of TMs in $L_2$. You will find you just cannot find a way. Now prove indeed you cannot. – Apass.Jack Mar 20 at 0:25
• Of course, if you can show that $L_2$ is not computably enumerable, then there is no such reduction. But could you please prove this? I really don't see why this is true. – CuriousKid7 Mar 20 at 0:31
• We can reduce $\overline{HALT}= \{⟨M, w⟩\mid\text{ TM }M\text{does not halt on input } w\}$ to $L_2$. – Apass.Jack Mar 20 at 0:54