TITLE: Why so many 'multi-part' definitions, as opposed to 'unified' ones?
QUESTION [8 upvotes]: Many definitions consist of multiple parts: an equivalence relation is symmetric AND reflexive AND transitive; a topology is closed over finite intersections AND over arbitrary unions; etc.  However, I've seen a number of cases where it seems simpler to combine the parts into a single definition: the result is often shorter and easier to calculate with.$
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\newcommand{\inf}[1]{\text{inf}(#1)}
\newcommand{\sup}[1]{\text{sup}(#1)}
\newcommand{\then}{\Rightarrow}
\newcommand{\when}{\Leftarrow}
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$\bullet\;$ As my most recent example, I discovered (through questions here on MSE) that $\;\inf{\cdots}\;$ can simply be defined by postulating $$
z \leq \inf{A} \;\equiv\; \langle \forall a : a \in A : z \leq a \rangle
$$ for any $\;z\;$ and lower-bounded $\;A\;$.  Contrast this with \begin{align}
& z \in A \;\then\; \inf{A} \leq z \\
& \langle \forall a : a \in A : z \leq a \rangle \;\then\; z \leq \inf{A} \\
\end{align} or even \begin{align}
& z \in A \;\then\; \inf{A} \leq z \\
& \langle \forall \epsilon : \epsilon > 0 : \langle \exists a : a \in A : a < \inf{A} + \epsilon \rangle \rangle \\
\end{align}
$\bullet\;$ For sets, the symmetric difference is often defined as $$
A \triangle B \;=\; (A \setminus B) \cup (B \setminus A)
$$ or $$
A \triangle B \;=\; (A \cup B) \setminus (A \cap B)
$$ while in practical proofs I find it much easier to work with $$
x \in A \triangle B \;\equiv\; x \in A \;\not\equiv\; x \in B
$$ for all $\;x\;$, since $\;\not\equiv\;$ is the logic-level equivalent of $\;\triangle\;$.
$\bullet\;$ The textbook definition of '$\;\mathscr T\text{ is a topology on }X\;$' is that \begin{align}
& \mathscr T \subseteq \mathscr P(X) \\
& \emptyset \in \mathscr T \\
& X \in \mathscr T \\
& \mathscr T\text{ is closed under }\cdots \cap \cdots \\
& \mathscr T\text{ is closed under }\bigcup \\
\end{align}  However, given closure under $\;\bigcup\;$, the first three conditions can be unified to just $$
\bigcup \mathscr T = X
$$ which has the very intuitive reading '$\;\mathscr T\;$ covers $\;X\;$'.
$\bullet\;$ In logic, I almost aways see the 'uniqueness quantifier' $\langle \exists! x :: P(x) \rangle$ ('there exists exactly one') defined as $$
\langle \exists x :: P(x) \rangle \;\land\; \langle \forall x,y : P(x) \land P(y) : x=y \rangle
$$ where $$
\langle \exists y :: \langle \forall x :: P(x) \;\equiv\; x = y \rangle \rangle
$$ is shorter and often seems much easier to work with.  And it has a nice symmetry: the $\;\then\;$ direction of the equivalence is uniqueness, which the $\;\when\;$ direction is existence.
$\bullet\;$ Finally, as an example from various domains, a statement of the form $\;P \equiv Q\;$ is very often seen as an invitation to give separate proofs for $\;P \then Q\;$ and $\;Q \then P\;$; and similarly for mutual inclusion for sets, and for proving equality of numbers using $\;\le\;$ and $\;\ge\;$, or even $\;\lt,=,\gt\;$.

The common pattern in all of the above, is that people seem to prefer 'multi-part' definitions over 'unified' definitions.  And I'm wondering why this is.
Does a proof which is split in parts perhaps have a proof-practical advantage?  As a kind of counterexample, a while ago I discovered that a relation $\;R\;$ on $\;A\;$ is an equivalence relation exactly when $$
aRb \:\equiv\: \langle \forall x :: aRx \equiv bRx\rangle
$$ holds for all $\;a,b\;$ (where $\;a,b,x\;$ range over $\;A\;$).  However, when I tried to actually use this definition to prove some relation to be an equivalence relation, then almost always the resulting proof was more complex than a proof of the three parts (reflexivity, symmetry, transitivity).  So in this specific example, the 'unified' definition did not really help me.  But in my experience, this has been the exception: 'unified' definitions almost always really work in practice for me.
Do the parts perhaps have an educational value?  Perhaps, at least initially, it is easier to build an intuition using separate parts, and then both those proofs and also later proofs are structured around that 'multi-part' intuition.
Is there perhaps an 'implicational bias'?  In other words, is it perhaps that I've been brought up in the 'school' of Dijkstra-Feijen, Gries-Schneider, et al., where there is an emphasis on equality and equivalence and symmetry, while most people approach proofs 'sequentially' based on inferences?
Or is something else at work here?

REPLY [2 votes]: Regarding binary relations, there are many important different types. An equivalence relation is symmetric, reflexive, and transitive. A linear order < is anti-symmetric, irreflexive, transitive, and satisfies trichotomy. A well-order is a linear order with an additional condition. A poset (the kind used in the set-theoretic topic called Forcing) is reflexive and transitive. And there are of course many others.  Instead of trying to compress the definitions, it is often more useful to list the parts, as it can then be seen how varying the parts results  in  other structures. 
Chess players say "To win you must use all your pieces". When attempting a proof, a list of properties, even if logically redundant, can help you to see some important property that you haven't used. 
Sometimes a defining list is easier to use because it incorporates more data: Let $\times$ be an associative binary operation on a set $G\ne \phi$ such that $\forall x,y\in G\;[\;(\exists! z\in G\;(x\times z=y)\land (\exists!z'\in G\; (z'\times x=y) \;].$ It takes some work to show that this meets all the "usual list" of conditions for a group. The usual def'n mentions an identity and unique two-sided inverses.
On the other hand, some writers do present def'ns that are much longer than most people would deem necessary.