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Theorem isros 30030
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 2687 . . . 4 (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆))
2 eleq2 2687 . . . . . . 7 (𝑠 = 𝑆 → ((𝑥𝑦) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑆))
3 eleq2 2687 . . . . . . 7 (𝑠 = 𝑆 → ((𝑥𝑦) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑆))
42, 3anbi12d 746 . . . . . 6 (𝑠 = 𝑆 → (((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠) ↔ ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)))
54raleqbi1dv 3138 . . . . 5 (𝑠 = 𝑆 → (∀𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠) ↔ ∀𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)))
65raleqbi1dv 3138 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠) ↔ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)))
71, 6anbi12d 746 . . 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 3352 . 2 (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆))))
10 3anass 1040 . 2 ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆))))
11 uneq1 3743 . . . . . 6 (𝑥 = 𝑢 → (𝑥𝑦) = (𝑢𝑦))
1211eleq1d 2683 . . . . 5 (𝑥 = 𝑢 → ((𝑥𝑦) ∈ 𝑆 ↔ (𝑢𝑦) ∈ 𝑆))
13 difeq1 3704 . . . . . 6 (𝑥 = 𝑢 → (𝑥𝑦) = (𝑢𝑦))
1413eleq1d 2683 . . . . 5 (𝑥 = 𝑢 → ((𝑥𝑦) ∈ 𝑆 ↔ (𝑢𝑦) ∈ 𝑆))
1512, 14anbi12d 746 . . . 4 (𝑥 = 𝑢 → (((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆) ↔ ((𝑢𝑦) ∈ 𝑆 ∧ (𝑢𝑦) ∈ 𝑆)))
16 uneq2 3744 . . . . . 6 (𝑦 = 𝑣 → (𝑢𝑦) = (𝑢𝑣))
1716eleq1d 2683 . . . . 5 (𝑦 = 𝑣 → ((𝑢𝑦) ∈ 𝑆 ↔ (𝑢𝑣) ∈ 𝑆))
18 difeq2 3705 . . . . . 6 (𝑦 = 𝑣 → (𝑢𝑦) = (𝑢𝑣))
1918eleq1d 2683 . . . . 5 (𝑦 = 𝑣 → ((𝑢𝑦) ∈ 𝑆 ↔ (𝑢𝑣) ∈ 𝑆))
2017, 19anbi12d 746 . . . 4 (𝑦 = 𝑣 → (((𝑢𝑦) ∈ 𝑆 ∧ (𝑢𝑦) ∈ 𝑆) ↔ ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
2115, 20cbvral2v 3170 . . 3 (∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆) ↔ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆))
22213anbi3i 1253 . 2 ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
239, 10, 223bitr2i 288 1 (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cdif 3556  cun 3557  c0 3896  𝒫 cpw 4135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564
This theorem is referenced by:  rossspw  30031  0elros  30032  unelros  30033  difelros  30034
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