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 ∈ 𝑉 → { 𝐴 } ∈ V )

Proof

Step	Hyp	Ref	Expression
1		sneq	⊢ ( 𝑥 = 𝐴 → { 𝑥 } = { 𝐴 } )
2		ax-bj-sn	⊢ ∀ 𝑥 ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥 )
3	2	spi	⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥 )
4		bj-axsn	⊢ ( { 𝑥 } ∈ V ↔ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥 ) )
5	3 4	mpbir	⊢ { 𝑥 } ∈ V
6	1 5	eqeltrrdi	⊢ ( 𝑥 = 𝐴 → { 𝐴 } ∈ V )
7	6	vtocleg	⊢ ( 𝐴 ∈ 𝑉 → { 𝐴 } ∈ V )