Singleton

Definition 1. The singleton of a set $a$ is

\[\{a\} = \{a, a\}.\]

Proposition 1. For any set $X$,

\[X \neq \{X\}.\]

Proof

If $X = \{X\}$, then $X \in X$, contrary to irreflexivity of $\in$.

References

[1]: Wikipedia contributors. Singleton (mathematics). Wikipedia, The Free Encyclopedia.
[2]: Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. p. 7.