The unordered pair $\{x,y\}$ and the singleton $\{x\} = \{x,x\}$

Let $x$ and $y$ be sets. The Axiom of Pairing: $\forall x\,\forall y\,\exists z\,\forall t\,(t \in z \leftrightarrow (t = x \vee t = y))$ gives a set whose elements are exactly $x$ and $y$, and The Axiom of Extensionality: $\forall x\,\forall y\,(\forall z\,(z \in x \leftrightarrow z \in y) \to x = y)$ shows there is only one such set; it is written $\{x,y\}$. Thus $\{x,y\}$ is the set whose elements are exactly $x$ and $y$, and $\{x\} := \{x,x\}$, the singleton of $x$, is the set whose only element is $x$:

$t \in \{x,y\} \leftrightarrow (t = x \vee t = y), \qquad t \in \{x\} \leftrightarrow t = x .$

• The pair is unordered. $t = x \vee t = y$ and $t = y \vee t = x$ are the same condition, so $\{x,y\} = \{y,x\}$, and the notation records no order. $\{a,b\} = \{c,d\}$ if and only if ($a = c$ and $b = d$) or ($a = d$ and $b = c$); in particular $\{a\} = \{c,d\}$ if and only if $a = c = d$ says exactly how much information a pair does carry, and The Kuratowski ordered pair $(a,b) := \{\{a\},\{a,b\}\}$ is the construction that recovers an order from unordered pairs alone.

• A singleton is not its element. $\{x\}$ has exactly one element, namely $x$, and $x$ and $\{x\}$ are different sets whenever they have different elements; $\{\varnothing\}$ has one element while $\varnothing$ has none, so the two are already distinct at the bottom of the hierarchy.