Decidability Doubt

Question

$L_1 = \{ \text{<M>}  | \ \text{M is a TM, } \text{M}_0  \ \text{is a TM that halts on all inputs and, } \text{M}_0 \in L(M) \}$

$L_2 = \{ \text{<M>}  | \ \text{M is a TM, } \text{M}_0  \ \text{is a TM that halts on all inputs and, } \text{M} \in L(M_0) \}$

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for $L_1$, $M$ accepts all encodings of TM which halts on all inputs. So this language is infinite because $\exists$ infinite number of TM's which accept $\Sigma^*$. 

Now $L(M)$ is not even RE. If that then how $L_1$ is RE.

Please explain both. 

Comments

for $L_2$ i understood that it is recursive because $M_0$ halts on every input and $M \in L(M_0)$ so $M_0$ will halt on all inputs of $L_2$ but unable to understand $\text{L}_1$