Theorem sspwuni

Description: Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006)

	Ref	Expression
Assertion	sspwuni	⊢ ( 𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		velpw	⊢ ( 𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵 )
2	1	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 )
3		dfss3	⊢ ( 𝐴 ⊆ 𝒫 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝐵 )
4		unissb	⊢ ( ∪ 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 )
5	2 3 4	3bitr4i	⊢ ( 𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵 )