Theorem snelpwg

Description: A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998) Put in closed form and avoid ax-nul . (Revised by BJ, 17-Jan-2025)

	Ref	Expression
Assertion	snelpwg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ { 𝐴 } ∈ 𝒫 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		snssg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ { 𝐴 } ⊆ 𝐵 ) )
2		snexg	⊢ ( 𝐴 ∈ 𝑉 → { 𝐴 } ∈ V )
3		elpwg	⊢ ( { 𝐴 } ∈ V → ( { 𝐴 } ∈ 𝒫 𝐵 ↔ { 𝐴 } ⊆ 𝐵 ) )
4	2 3	syl	⊢ ( 𝐴 ∈ 𝑉 → ( { 𝐴 } ∈ 𝒫 𝐵 ↔ { 𝐴 } ⊆ 𝐵 ) )
5	1 4	bitr4d	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ { 𝐴 } ∈ 𝒫 𝐵 ) )