Theorem expanduniss 41536

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 4843	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
2		dfss2 3877	. . 3 ⊢ (𝑥 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
3	2	expandral 41533	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))
4	1, 3	bitri 278	1 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541   ∈ wcel 2110  ∀wral 3054   ⊆ wss 3857  ∪ cuni 4809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2706

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-ral 3059  df-v 3403  df-in 3864  df-ss 3874  df-uni 4810

This theorem is referenced by:  ismnuprim  41537