A pair of identical elements is a singleton

Statement

Let $t$ be a set. By the axiom of pairing we may construct a unique (unordered) pair, which up until now we have denoted by $\{t,t\}$. We now show that $\{t,t\}$ is a singleton, thus justifying the notation:

• $\{t\}$ for a pair consisting of the same thing for both parts.

Formally we must show:

• $\exists x[x\in\{t,t\}\rightarrow\forall y(y\in\{t,t\}\rightarrow y\eq x)]$ (as per definition of singleton

Proof of claim

[Expand]

Recall the definition: for singleton

Let $X$ be a set. We call $X$ a singleton if:

• $\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]$^Caveat:See:
  • In words: $X$ is a singleton if: there exists a thing such that ( the thing is in $X$ and for any stuff ( if that stuff is in $X$ then the stuff is the thing ) )

More concisely this may be written:

• $\exists t\in X\forall s\in X[t\eq s]$

TODO: When the paring axiom has a page, do the same thing

• Template:M\forall A\forall B\exists C\forall x(x\in C\leftrightarrow x=A\vee x=B) this is the pairing axiom, in this case $A$ and $B$ are $t$ and $C$ is the (it turns out unique) set $\{t,t\}$

Proof body

• Choose $x:\eq t$

TODO: This is wrong, saving work and switching computer

• • $x\in\{t,t\}$ as a result.

Warwick lecture notes - Set Theory - 2011 - Adam Epstein - page 2.75.