Theorem snssg

Description: The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of Quine p. 49. (Contributed by NM, 22-Jul-2001) (Proof shortened by BJ, 1-Jan-2025)

	Ref	Expression
Assertion	snssg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ { 𝐴 } ⊆ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		snssb	⊢ ( { 𝐴 } ⊆ 𝐵 ↔ ( 𝐴 ∈ V → 𝐴 ∈ 𝐵 ) )
2	1	bicomi	⊢ ( ( 𝐴 ∈ V → 𝐴 ∈ 𝐵 ) ↔ { 𝐴 } ⊆ 𝐵 )
3		elex	⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V )
4		imbibi	⊢ ( ( ( 𝐴 ∈ V → 𝐴 ∈ 𝐵 ) ↔ { 𝐴 } ⊆ 𝐵 ) → ( 𝐴 ∈ V → ( 𝐴 ∈ 𝐵 ↔ { 𝐴 } ⊆ 𝐵 ) ) )
5	2 3 4	mpsyl	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ { 𝐴 } ⊆ 𝐵 ) )