Theorem snelpwi

Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by Alan Sare, 25-Aug-2011)

	Ref	Expression
Assertion	snelpwi	⊢ ( 𝐴 ∈ 𝐵 → { 𝐴 } ∈ 𝒫 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		snssi	⊢ ( 𝐴 ∈ 𝐵 → { 𝐴 } ⊆ 𝐵 )
2		snex	⊢ { 𝐴 } ∈ V
3	2	elpw	⊢ ( { 𝐴 } ∈ 𝒫 𝐵 ↔ { 𝐴 } ⊆ 𝐵 )
4	1 3	sylibr	⊢ ( 𝐴 ∈ 𝐵 → { 𝐴 } ∈ 𝒫 𝐵 )