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Metamath Proof Explorer

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 ∈ 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 )