In Jech's third millennium, theorem 13.9 states that transitive class $M$ is an inner model of $ZF$ if and only if it is closed under Godel operations and is almost universal. The proof of first part includes the following.
If $X$ is a subset of an inner model $M$, then $X ⊂ V_α ∩ M$ for some $α$, and $V_α ∩ M\in M$ because $α ∈ M$ and $V_α ∩M = V^M_α$.
I think the proof is problematic because if $X$ contains all ordinals (it is possible since $M$ does), then there is no $\alpha$ that $X ⊂ V_α ∩ M$ for some $α$, and $V_α ∩ M\in M$.
Notice that it says $X$ is a subset of $M$. That means $X$ is a set, not just a class, so it cannot contain all the ordinals, and is contained in some $V_\alpha$ since every set is contained in some $V_\alpha$.