nLab
subset

In traditional set theory, a subset of a set $A$ is a set $B$ with the property that

$$x\in B\Rightarrow x\in A$$x \in B \;\Rightarrow\; x \in A

for any object $x$ whatsoever. One writes $B\subseteq A$ or $B\subset A$ (depending on the author) if $B$ has this property. Set theory's axiom of extensionality says that $A=B$ if (and only if) $A\subseteq B$ and $B\subseteq A$.

In other foundations of mathematics, this definition doesn't make sense, because there is no global membership predicate $\in$ (and there may not be a global equality predicate either). Accordingly, we define a subset of $A$ to be a set $B$ with the structure of an injection

$$i:B\hookrightarrow A.$$i: B \hookrightarrow A.

We can still define a local membership predicate ${\in }_A$ as follows: Given an element $x$ of $A$ and a subset $B$ (technically, $(B,i)$) of $A$,

(1)$$x{\in }_{A}B\iff \exists (y:B),x=i(y).$$x \in_A B \;\Leftrightarrow\; \exists(y: B),\; x = i(y).

Similarly, we can define a local containment predicate ${\subseteq }_A$ as follows: Given subsets $B$ and $C$ of $A$,

$$B{\subseteq }_{A}C\iff \forall (x:A),x\in B\Rightarrow x\in C.$$B \subseteq_A C \;\Leftrightarrow\; \forall(x: A),\; x \in B \;\Rightarrow\; x \in C.

We can also define a local equality predicate ${=}_A$ on subsets of $A$:

$$B{=}_{A}C\iff B\subseteq C\wedge C\subseteq B.$$B =_A C \;\Leftrightarrow\; B \subseteq C \;\wedge\; C \subseteq B.

In foundations that already have a global equality predicate on sets (and functions between equal sets), this local equality predicate must be interpreted as an equivalence relation; then a subset of $A$ is technically an equivalence class of injections to $A$ rather than simply an injection to $A$.

As you can see from the right-hand sides of the above sequence of definitions, one usually suppresses the subscript $A$ from the notation. Even the right-hand side of (1) may use a local equality relation ${=}_A$ on elements of $A$ (which is not the same as the local equality relation on subsets of $A$, of course). It may be necessary to distinguish $x:E$ (introducing a variable $x$ for an element of a given set $E$) from $x{\in }_{A}E$ (the proposition that a given element $x$ of a given set $A$ is a member of a given subset $E$ of $A$). Some authors may use $x\in A$ for either or both of these, trusting on context (particularly whether $x$ has been introduced before) to distinguish them. Another notational convenience is to suppress the injection $i$, writing $y$ instead of $i(y)$.

The concept of subset as it appears here generalises to subobject in category theory. To be precise, a subset of $A$ is exactly a subobject of $A$ as an object of the category Set.