Theorem pwssb

Description: Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006)

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

Proof

Step	Hyp	Ref	Expression
1		sspwuni	⊢ ( 𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵 )
2		unissb	⊢ ( ∪ 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 )
3	1 2	bitri	⊢ ( 𝐴 ⊆ 𝒫 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 )