Theorem pwss

Description: Subclass relationship for power class. (Contributed by NM, 21-Jun-2009)

	Ref	Expression
Assertion	pwss	⊢ ( 𝒫 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ss	⊢ ( 𝒫 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵 ) )
2		velpw	⊢ ( 𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴 )
3	2	imbi1i	⊢ ( ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵 ) ↔ ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵 ) )
4	3	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝐵 ) ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵 ) )
5	1 4	bitri	⊢ ( 𝒫 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐵 ) )