Theorem elpwinss

Description: An element of the powerset of B intersected with anything, is a subset of B . (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Assertion	elpwinss	⊢ ( 𝐴 ∈ ( 𝒫 𝐵 ∩ 𝐶 ) → 𝐴 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elinel1	⊢ ( 𝐴 ∈ ( 𝒫 𝐵 ∩ 𝐶 ) → 𝐴 ∈ 𝒫 𝐵 )
2	1	elpwid	⊢ ( 𝐴 ∈ ( 𝒫 𝐵 ∩ 𝐶 ) → 𝐴 ⊆ 𝐵 )