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 $\in$.
Pub Feb 16, 2026 Up Jun 23, 2026
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 $\in$.