User:Ascii/Prose Test/Set Theory

A set is a subset of itself: $\forall S: S \subseteq S$. The singleton of an element of a set is a subset of that set: $x \in S \iff \{x\} \subseteq S$. The subset relation $\subseteq$ is transitive: $\paren {R \subseteq S} \land \paren {S \subseteq T} \implies R \subseteq T$. There are equivalent definitions of set equality: $S = T \iff \paren {\forall x: x \in S \iff x \in T}$ and $S = T \iff S \subseteq T \land T \subseteq S$.