I'm trying to understand proofs for set theory, and need some help representing my reasoning for a proof. This is super basic but I want to make sure I definitely understand this before I move on.
Given
$x\in A$
$A\subseteq B$
Prove
$x\in B$
So $A\subseteq B$ is the same as saying $\forall n(n\in A \rightarrow n\in B )$
Is it safe to say that $x$ can be an instance of $n$, if $n$ is all elements in the universe of discourse, and do I need to specify that somehow in a proof? Do I need to specify that $x$ and $n$ are in the same universe or is that implied?
The fact that $x \in A$ means that $x$ and $A$ must be in the same universe of discourse, so no specification is needed. Also, in general, the universe of discourse never changes within a single series of statements.
Finally, in set theory, the universe of discourse is usually everything. When we say $A \subseteq B$, we don't mean "$A$ is a subset of $B$ as far as the present universe of discourse is concerned", we mean "literally every element of $A$ is also an element of $B$".