Theorem expanduniss 44289

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem expanduniss

Step	Hyp	Ref	Expression
1		unissb 4906	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
2		df-ss 3934	. . 3 ⊢ (𝑥 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
3	2	expandral 44286	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
4	1, 3	bitri 275	1 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   ∈ wcel 2109  ∀wral 3045   ⊆ wss 3917  ∪ cuni 4874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-v 3452  df-ss 3934  df-uni 4875

This theorem is referenced by:  ismnuprim  44290