Theorem bj-axsn

Description: Two ways of stating the axiom of singleton (which is the universal closure of either side, see ax-bj-sn ). (Contributed by BJ, 12-Jan-2025) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-axsn	⊢ ( { 𝑥 } ∈ V ↔ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		velsn	⊢ ( 𝑧 ∈ { 𝑥 } ↔ 𝑧 = 𝑥 )
2	1	bj-clex	⊢ ( { 𝑥 } ∈ V ↔ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ 𝑧 = 𝑥 ) )