Theorem snexgALT

Description: Alternate proof of snexg based on vsnex , which uses an instance of ax-sep . (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) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	snexgALT	⊢ ( 𝐴 ∈ 𝑉 → { 𝐴 } ∈ V )

Proof

Step	Hyp	Ref	Expression
1		sneq	⊢ ( 𝑥 = 𝐴 → { 𝑥 } = { 𝐴 } )
2		vsnex	⊢ { 𝑥 } ∈ V
3	1 2	eqeltrrdi	⊢ ( 𝑥 = 𝐴 → { 𝐴 } ∈ V )
4	3	vtocleg	⊢ ( 𝐴 ∈ 𝑉 → { 𝐴 } ∈ V )