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 e. V -> { A } e. _V )

Proof

Step	Hyp	Ref	Expression
1		sneq	|- ( x = A -> { x } = { A } )
2		vsnex	|- { x } e. _V
3	1 2	eqeltrrdi	|- ( x = A -> { A } e. _V )
4	3	vtocleg	|- ( A e. V -> { A } e. _V )