Theorem expanduniss

Description: Expand U. A C_ B to primitives. (Contributed by Rohan Ridenour, 13-Aug-2023)

	Ref	Expression
Assertion	expanduniss	⊢ ( ∪ 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		unissb	⊢ ( ∪ 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 )
2		dfss2	⊢ ( 𝑥 ⊆ 𝐵 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵 ) )
3	2	expandral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵 ) ) )
4	1 3	bitri	⊢ ( ∪ 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵 ) ) )