View source for Singleton (set theory)

__TOC__
=={{subpage|Definition}}==
{{/Definition}}<!--
-->** For proof see '''Claim 1'''.

==Significance==
Notice that we have manage to define a set containing one thing without any notion of the number 1. 
==See next==
* [[A pair of identical elements is a singleton]]
==Proof of claims==
# {{M|\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X\forall s\in X[t\eq s]\big)}}
#* By "''[[rewriting for-all and exists within set theory]]''" we see:
#** {{M|\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X[\forall s(s\in X\rightarrow s\eq t)]\big)}}
#**: {{M|\iff\big(\exists t\in X\forall s(s\in X\rightarrow s\eq t)\big)}} by simplification
#**: {{M|\iff\big(\exists t\in X\forall s[s\in X\rightarrow s\eq t]\big)}} by changing the bracket style
#**: {{M|\iff\big(\exists t\in X\forall s\in X[s\eq t]\big)}} by re-writing again.
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Elementary Set Theory|Set Theory}}