I'm reading Jech's Axiom of Choice, and when he makes his case that the constructible universe L is a model of ZFC, he says it's "easy to see" that it's transitive and almost universal. I see transitive, but how is it almost universal? (That is, every subset that's a set is also subset of an element.)
For reference, his definition of L is the union across all ordinals of, for non-limit ordinals, the intersection of the predecessor's power set and the same's closure under Gödel operations, and for limit ordinals, the union of all values within.
Your definition of almost universal is wrong! For $L$ to be almost universal means that if $A$ is any set with $A\subseteq L$, then there is some set $B\in L$ such that $A\subseteq B$ (but $A$ itself may not be in $L$).
To prove this, just note that for each $a\in A$, there is some least ordinal $\alpha_a$ such that $a\in L_{\alpha_a}$. If $\beta$ is the supremum of all these ordinals, then $a\in L_\beta$ for all $a\in A$. So $A\subseteq L_\beta$, and since $L_\beta\in L$ this means we can take $B=L_\beta$.