Theorem isros 32136

Description: The property of being a rings of sets, i.e. containing the empty set, and closed under finite union and set complement. (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
isros	⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)))

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

Allowed substitution hints:   𝑄(𝑥,𝑦,𝑣,𝑢,𝑠)   𝑂(𝑥,𝑦,𝑣,𝑢)

Proof of Theorem isros

Step	Hyp	Ref	Expression
1		eleq2 2827	. . . 4 ⊢ (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆))
2		eleq2 2827	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑥 ∪ 𝑦) ∈ 𝑠 ↔ (𝑥 ∪ 𝑦) ∈ 𝑆))
3		eleq2 2827	. . . . . . 7 ⊢ (𝑠 = 𝑆 → ((𝑥 ∖ 𝑦) ∈ 𝑠 ↔ (𝑥 ∖ 𝑦) ∈ 𝑆))
4	2, 3	anbi12d 631	. . . . . 6 ⊢ (𝑠 = 𝑆 → (((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠) ↔ ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆)))
5	4	raleqbi1dv 3340	. . . . 5 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠) ↔ ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆)))
6	5	raleqbi1dv 3340	. . . 4 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆)))
7	1, 6	anbi12d 631	. . 3 ⊢ (𝑠 = 𝑆 → ((∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆))))
8		isros.1	. . 3 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}
9	7, 8	elrab2 3627	. 2 ⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆))))
10		3anass 1094	. 2 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆)) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆))))
11		uneq1 4090	. . . . . 6 ⊢ (𝑥 = 𝑢 → (𝑥 ∪ 𝑦) = (𝑢 ∪ 𝑦))
12	11	eleq1d 2823	. . . . 5 ⊢ (𝑥 = 𝑢 → ((𝑥 ∪ 𝑦) ∈ 𝑆 ↔ (𝑢 ∪ 𝑦) ∈ 𝑆))
13		difeq1 4050	. . . . . 6 ⊢ (𝑥 = 𝑢 → (𝑥 ∖ 𝑦) = (𝑢 ∖ 𝑦))
14	13	eleq1d 2823	. . . . 5 ⊢ (𝑥 = 𝑢 → ((𝑥 ∖ 𝑦) ∈ 𝑆 ↔ (𝑢 ∖ 𝑦) ∈ 𝑆))
15	12, 14	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝑢 → (((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆) ↔ ((𝑢 ∪ 𝑦) ∈ 𝑆 ∧ (𝑢 ∖ 𝑦) ∈ 𝑆)))
16		uneq2 4091	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑢 ∪ 𝑦) = (𝑢 ∪ 𝑣))
17	16	eleq1d 2823	. . . . 5 ⊢ (𝑦 = 𝑣 → ((𝑢 ∪ 𝑦) ∈ 𝑆 ↔ (𝑢 ∪ 𝑣) ∈ 𝑆))
18		difeq2 4051	. . . . . 6 ⊢ (𝑦 = 𝑣 → (𝑢 ∖ 𝑦) = (𝑢 ∖ 𝑣))
19	18	eleq1d 2823	. . . . 5 ⊢ (𝑦 = 𝑣 → ((𝑢 ∖ 𝑦) ∈ 𝑆 ↔ (𝑢 ∖ 𝑣) ∈ 𝑆))
20	17, 19	anbi12d 631	. . . 4 ⊢ (𝑦 = 𝑣 → (((𝑢 ∪ 𝑦) ∈ 𝑆 ∧ (𝑢 ∖ 𝑦) ∈ 𝑆) ↔ ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)))
21	15, 20	cbvral2vw 3396	. . 3 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆) ↔ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆))
22	21	3anbi3i 1158	. 2 ⊢ ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∪ 𝑦) ∈ 𝑆 ∧ (𝑥 ∖ 𝑦) ∈ 𝑆)) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)))
23	9, 10, 22	3bitr2i 299	1 ⊢ (𝑆 ∈ 𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢 ∈ 𝑆 ∀𝑣 ∈ 𝑆 ((𝑢 ∪ 𝑣) ∈ 𝑆 ∧ (𝑢 ∖ 𝑣) ∈ 𝑆)))

Syntax hints:   ↔ wb 205   ∧ 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:  rossspw  32137  0elros  32138  unelros  32139  difelros  32140