Theorem bj-axsn

Description: Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn ). (Contributed by BJ, 12-Jan-2025) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-axsn	$$⊢\left\{x\right\}\in \mathrm{V}\leftrightarrow \exists y\forall z\left(z\in y\leftrightarrow z=x\right)$$

Proof

Step	Hyp	Ref	Expression
1		velsn	$$⊢z\in \left\{x\right\}\leftrightarrow z=x$$
2	1	bj-clex	$$⊢\left\{x\right\}\in \mathrm{V}\leftrightarrow \exists y\forall z\left(z\in y\leftrightarrow z=x\right)$$