Set Notation - Inclusion and Proper Inclusion By definition:

If$ \ A$ and $B\ $are sets and every element of$ \ A\ $ is also an element of $ \ B \ $then we can say $ \ A\ $is a subset of $ \ B, \ $and denote this by $ \ A\subset B\ $or $ \ A\subseteq B.\ $Or, equivalently, we can say that$ \ B$ is a super set of $\ A \ $(if every element of A
   is also an element of B), which is denoted by $ \ B\supseteq\ A.\ $If $ \ A \ $ is $\neq\ B \ $(there exists at least one element of $ \ B \ $which is not an element of $ \ A), \ $then $ \ A \ $is a proper subset of $ \ B \ $, which is denoted$ \ A\subsetneq\ B. \ $ Or, equivalently, $ \ B$ is a proper super set of $\ A \  $(if $A\subset B,$ but $A\neq B$) which is denoted by $ \ B\supsetneq\ A.\ $

In the case where where $ \ A \ $ is $\neq\ B, \ $but every element of $A$ is also an element of $B,$ are $ \ A\subset B,\ A\subseteq B,\ B\supseteq A,\ A\subsetneq B,\ B\supsetneq A \ $all equivalent? In the case where where $ \ A=B \ $are $\ A\subset B,\ B\subset A,\ A\subseteq B,\ B\supseteq A,\ A\supseteq B,\ B\subseteq A\ $all equivalent? It's kind of strange to think that if two sets are equal to one another, that they are also both supersets of each other. It seems more "natural," to me at least, that the whole superset "concept," where $ \ A\supseteq B\ $or $ \ B\supseteq A,\ $should be reserved for when $ \ B\neq A\ $(but every element of $B$ is in $A$) or $ \ A\neq B \ $(but every element of $A$ is in $B$). Of course in both of these cases we would use the proper superset notation. 
 A: In fact, one way to prove that two sets are equal is to show that they are both subsets/supersets of each other, i.e. $A = B \iff \left( A\subset B \right)\wedge \left(B\subset A\right)$.
The 'equivalencies' you've written are not exactly the way you are thinking. It's true that if $A$ is a subset of $B$ but not equal to $B$ then $A\subset B$, $A\subseteq B$, $B\supseteq A$, $B\supset A$ are all true but it is sloppy. The most correct thing to write in this case is $A\subsetneq B$ or $B\supsetneq A$. Think about the analogous concepts with numbers, of course $2+2=4$ but we can be sloppy and write $2+2\geq 4$ or $4\leq 2+2$. The proper subset/supset notation is more precise, but if you have a proper subset/supset then yes, you also have subset/supset inclusion as well.
A: Note: some authors use $\subset$ and $\varsubsetneq$ interchangeably whereas others use $\subset$ and $\subseteq$ interchangeably.
In the case that $A$ is a proper subset of $B$, you can still write $A \subseteq B$, but this is the same as writing $2\le 3$ instead of $2 < 3$ — both statements are true, but one is more conclusive.
