Theorem unelros 32139

Description: A ring of sets is closed under union. (Contributed by Thierry Arnoux, 18-Jul-2020.)

Hypothesis

Ref	Expression
isros.1	⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}

Assertion

Ref	Expression
unelros	⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∪ 𝐵) ∈ 𝑆)

Distinct variable groups:   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑠)   𝐵(𝑥,𝑦,𝑠)   𝑄(𝑥,𝑦,𝑠)   𝑂(𝑥,𝑦)

Proof of Theorem unelros

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1136	. . 3 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆)
2		simp3 1137	. . 3 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐵 ∈ 𝑆)
3		isros.1	. . . . . 6 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}
4	3	isros 32136	. . . . 5 ⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)))
5	4	simp3bi 1146	. . . 4 ⊢ (𝑆 ∈ 𝑄 → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆))
6	5	3ad2ant1 1132	. . 3 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆))
7		uneq1 4090	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑢 ∪ 𝑣) = (𝐴 ∪ 𝑣))
8	7	eleq1d 2823	. . . . 5 ⊢ (𝑢 = 𝐴 → ((𝑢 ∪ 𝑣) ∈ 𝑆 ↔ (𝐴 ∪ 𝑣) ∈ 𝑆))
9		difeq1 4050	. . . . . 6 ⊢ (𝑢 = 𝐴 → (𝑢 ∖ 𝑣) = (𝐴 ∖ 𝑣))
10	9	eleq1d 2823	. . . . 5 ⊢ (𝑢 = 𝐴 → ((𝑢 ∖ 𝑣) ∈ 𝑆 ↔ (𝐴 ∖ 𝑣) ∈ 𝑆))
11	8, 10	anbi12d 631	. . . 4 ⊢ (𝑢 = 𝐴 → (((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆) ↔ ((𝐴 ∪ 𝑣) ∈ 𝑆 ∧ (𝐴 ∖ 𝑣) ∈ 𝑆)))
12		uneq2 4091	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝐴 ∪ 𝑣) = (𝐴 ∪ 𝐵))
13	12	eleq1d 2823	. . . . 5 ⊢ (𝑣 = 𝐵 → ((𝐴 ∪ 𝑣) ∈ 𝑆 ↔ (𝐴 ∪ 𝐵) ∈ 𝑆))
14		difeq2 4051	. . . . . 6 ⊢ (𝑣 = 𝐵 → (𝐴 ∖ 𝑣) = (𝐴 ∖ 𝐵))
15	14	eleq1d 2823	. . . . 5 ⊢ (𝑣 = 𝐵 → ((𝐴 ∖ 𝑣) ∈ 𝑆 ↔ (𝐴 ∖ 𝐵) ∈ 𝑆))
16	13, 15	anbi12d 631	. . . 4 ⊢ (𝑣 = 𝐵 → (((𝐴 ∪ 𝑣) ∈ 𝑆 ∧ (𝐴 ∖ 𝑣) ∈ 𝑆) ↔ ((𝐴 ∪ 𝐵) ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆)))
17	11, 16	rspc2va 3571	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)) → ((𝐴 ∪ 𝐵) ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆))
18	1, 2, 6, 17	syl21anc 835	. 2 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴 ∪ 𝐵) ∈ 𝑆 ∧ (𝐴 ∖ 𝐵) ∈ 𝑆))
19	18	simpld 495	1 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∪ 𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   ∖ cdif 3884   ∪ cun 3885  ∅c0 4256  𝒫 cpw 4533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892

This theorem is referenced by:  fiunelros  32142