Treating Relations as Sets I've been learning about relations and so far and I think I understand the basics just fine.
If $R$ is a relation from $A$ to $B$, then $R \subseteq A \times B$. The ordered pairs in $R$ define a relation between the objects in A and B, so $a \in A$ is related to $b \in B$ by $R$ if $(a, b) \in R$.
I'm also fine with notions such as 'relations on a set', where $R \subseteq A \times A$. I'm also fine with concepts such as reflexive, symmetric and transitive relations.
I can wrap my head around treating the symbol $\leq$ as a set. For example, we define $\mathbb{N} = \{0, 1, 2, ...\}$, and we define conditions under which $x \leq y$, for all $x, y \in \mathbb{N}$. I suppose it'd be something like:
$\leq \, = \{ (x, y) \in \mathbb{N} \times \mathbb{N} \text{ | } y = x + d \text{ for some } d \in \mathbb{N} \}$
We can then construct $\mathbb{Z}$ and overload the $\leq$ symbol for the ordering relation in the set of integers, and the process repeats for $\mathbb{Q}$ and $\mathbb{R}$. We can then show $\leq$ is reflexive, anti-symmetric and transitive, and everything works out just fine.
The process above is what I'm comfortable with. However, I'm having a bit of trouble wrapping my head around $\subseteq$, $\in$ and $=$.
Relations like $\leq$ make sense for me because we start from sets, and then we define the reals, and then further define $\leq \, \subseteq \mathbb{R} \times \mathbb{R}$ (depending on the construction method $\leq$ might be defined differently but the end result is still the same). After the stage is set, we just proceed using $\leq$ as I was taught in secondary school with no trouble.
However, for $\subseteq$, I see books are referring to this symbol as a relation, but it seems like the situation is entirely different. If the domain of $\leq$ is the set of reals, then what is the domain of $\subseteq$? The set of all sets? Isn't this set not allowed to exist in ZF? If the domain is not the set of all sets, then do we literally have to define a new domain based on the sets we are comparing every time we use $\subseteq$? It's not like it's impossible, and it's not like these concerns will affect the actual process of doing math, but I just find that technically such a cumbersome process is required when we use $\subseteq$ is a bit unsatisfying.
My concerns for $\in$ and $=$ are also similar. It seems like this problem arises because $\subseteq$, $\in$ and $=$ are used to express relationships between sets, unlike $\leq$ which is designed to operate in the little pocket universe of $\mathbb{R}$ which we have defined for ourselves.
(Thanks in advance for taking the time to go through my long question. I'm a physics student who has recently acquired a taste for more rigorous mathematics, and so far I know about mathematical logic, sets and things like the Peano axioms at the level of an introductory course for undergraduates. At this point, I think I understand the ZF axioms so I should be able to comprehend any answer that might touch upon these concepts. I am currently reading Mathematical Proofs: A Transition to Advanced Mathematics and a bit of Tao's Analysis I.)
 A: 
If the domain of ≤ is the set of reals, then what is the domain of ⊆? The set of all sets? Isn't this set not allowed to exist in ZF?

Actually, as you know a relation $R$ from $A$ to $B$ is a subset of $A\times B$. When we consider a relation we have to mention the domain and codomain.
When you say $\subseteq$  is a relation it does not make any sense unless you specify $A$ and $B$.
Given a set $X$ you can define the relation $\subseteq$ as a subset of $\mathsf{P}(X) \times \mathsf{P}(X)$ where $\mathsf{P}(X)$ is the power set of $X$. Most probably in textbooks if they consider subsets of real numbers then we can take $X= \mathbb{R}$ or generally they start like "Consider a set $A$ and let $B, C$ be subsets of $A$ then $\ldots$" and in this case, we take $X=A$.
So there does not arise a situation considering the set of sets.
Note that we can define $\le$ on the sets other than real numbers also. For example consider $S= \{z\in \mathbb{C} \mid \Re(z) = 0 \}$ and define a relation $\le$ as $z_1 \le z_2$ iff $\Im(z_1) \le \Im(z_2)$ for $z_1,z_2\in S$. This relation is similar to the relation $\le$ on $\mathbb{R}$ so when we mention a relation we must specify the underlying set.
