In the usual development of set theory, the expression $A\subseteq B$ is defined to mean $\forall x(x\in A\to x\in B)$.
Similarly, $A\cup B$ denotes the set of all $x$ satisfying $x\in A\lor x\in B$, and $A\cap B$ denotes the set of all $x$ satisfying $x\in A\land x\in B$. The set operations are in a sense just shorthand for the underlying logical operations.