Theorem snexg

Description: A singleton built on a set is a set. Special case of snex which does not require ax-nul and is intuitionistically valid. (Contributed by NM, 7-Aug-1994) (Revised by Mario Carneiro, 19-May-2013) Extract from snex and shorten proof. (Revised by BJ, 15-Jan-2025)

	Ref	Expression
Assertion	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		vsnex	$$⊢\left\{x\right\}\in \mathrm{V}$$
3	1 2	eqeltrrdi	$$⊢x=A\to \left\{A\right\}\in \mathrm{V}$$
4	3	vtocleg	$$⊢A\in V\to \left\{A\right\}\in \mathrm{V}$$