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Theorem isros 29558
 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 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
Assertion
Ref Expression
isros (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 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
StepHypRef Expression
1 eleq2 2677 . . . 4 (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆))
2 eleq2 2677 . . . . . . 7 (𝑠 = 𝑆 → ((𝑥𝑦) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑆))
3 eleq2 2677 . . . . . . 7 (𝑠 = 𝑆 → ((𝑥𝑦) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑆))
42, 3anbi12d 743 . . . . . 6 (𝑠 = 𝑆 → (((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠) ↔ ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)))
54raleqbi1dv 3123 . . . . 5 (𝑠 = 𝑆 → (∀𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠) ↔ ∀𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)))
65raleqbi1dv 3123 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠) ↔ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)))
71, 6anbi12d 743 . . 3 (𝑠 = 𝑆 → ((∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆))))
8 isros.1 . . 3 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
97, 8elrab2 3333 . 2 (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆))))
10 3anass 1035 . 2 ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆))))
11 uneq1 3722 . . . . . 6 (𝑥 = 𝑢 → (𝑥𝑦) = (𝑢𝑦))
1211eleq1d 2672 . . . . 5 (𝑥 = 𝑢 → ((𝑥𝑦) ∈ 𝑆 ↔ (𝑢𝑦) ∈ 𝑆))
13 difeq1 3683 . . . . . 6 (𝑥 = 𝑢 → (𝑥𝑦) = (𝑢𝑦))
1413eleq1d 2672 . . . . 5 (𝑥 = 𝑢 → ((𝑥𝑦) ∈ 𝑆 ↔ (𝑢𝑦) ∈ 𝑆))
1512, 14anbi12d 743 . . . 4 (𝑥 = 𝑢 → (((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆) ↔ ((𝑢𝑦) ∈ 𝑆 ∧ (𝑢𝑦) ∈ 𝑆)))
16 uneq2 3723 . . . . . 6 (𝑦 = 𝑣 → (𝑢𝑦) = (𝑢𝑣))
1716eleq1d 2672 . . . . 5 (𝑦 = 𝑣 → ((𝑢𝑦) ∈ 𝑆 ↔ (𝑢𝑣) ∈ 𝑆))
18 difeq2 3684 . . . . . 6 (𝑦 = 𝑣 → (𝑢𝑦) = (𝑢𝑣))
1918eleq1d 2672 . . . . 5 (𝑦 = 𝑣 → ((𝑢𝑦) ∈ 𝑆 ↔ (𝑢𝑣) ∈ 𝑆))
2017, 19anbi12d 743 . . . 4 (𝑦 = 𝑣 → (((𝑢𝑦) ∈ 𝑆 ∧ (𝑢𝑦) ∈ 𝑆) ↔ ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
2115, 20cbvral2v 3155 . . 3 (∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆) ↔ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆))
22213anbi3i 1248 . 2 ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
239, 10, 223bitr2i 287 1 (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ∖ cdif 3537   ∪ cun 3538  ∅c0 3874  𝒫 cpw 4108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545 This theorem is referenced by:  rossspw  29559  0elros  29560  unelros  29561  difelros  29562
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