Theorem bj-snexg

Description: A singleton built on a set is a set. Contrary to bj-snex , this proof is intuitionistically valid and does not require ax-nul . (Contributed by NM, 7-Aug-1994) Extract it from snex and prove it from ax-bj-sn . (Revised by BJ, 12-Jan-2025) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-snexg	$$⊢A\in V\to \left\{A\right\}\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		sneq	$$⊢x=A\to \left\{x\right\}=\left\{A\right\}$$
2		ax-bj-sn	$$⊢\forall x\exists y\forall z\left(z\in y\leftrightarrow z=x\right)$$
3	2	spi	$$⊢\exists y\forall z\left(z\in y\leftrightarrow z=x\right)$$
4		bj-axsn	$$⊢\left\{x\right\}\in \mathrm{V}\leftrightarrow \exists y\forall z\left(z\in y\leftrightarrow z=x\right)$$
5	3 4	mpbir	$$⊢\left\{x\right\}\in \mathrm{V}$$
6	1 5	eqeltrrdi	$$⊢x=A\to \left\{A\right\}\in \mathrm{V}$$
7	6	vtocleg	$$⊢A\in V\to \left\{A\right\}\in \mathrm{V}$$