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Theorem isros 31501
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 2902 . . . 4 (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆))
2 eleq2 2902 . . . . . . 7 (𝑠 = 𝑆 → ((𝑥𝑦) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑆))
3 eleq2 2902 . . . . . . 7 (𝑠 = 𝑆 → ((𝑥𝑦) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑆))
42, 3anbi12d 633 . . . . . 6 (𝑠 = 𝑆 → (((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠) ↔ ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)))
54raleqbi1dv 3384 . . . . 5 (𝑠 = 𝑆 → (∀𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠) ↔ ∀𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)))
65raleqbi1dv 3384 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠) ↔ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)))
71, 6anbi12d 633 . . 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 3658 . 2 (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆))))
10 3anass 1092 . 2 ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆))))
11 uneq1 4107 . . . . . 6 (𝑥 = 𝑢 → (𝑥𝑦) = (𝑢𝑦))
1211eleq1d 2898 . . . . 5 (𝑥 = 𝑢 → ((𝑥𝑦) ∈ 𝑆 ↔ (𝑢𝑦) ∈ 𝑆))
13 difeq1 4067 . . . . . 6 (𝑥 = 𝑢 → (𝑥𝑦) = (𝑢𝑦))
1413eleq1d 2898 . . . . 5 (𝑥 = 𝑢 → ((𝑥𝑦) ∈ 𝑆 ↔ (𝑢𝑦) ∈ 𝑆))
1512, 14anbi12d 633 . . . 4 (𝑥 = 𝑢 → (((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆) ↔ ((𝑢𝑦) ∈ 𝑆 ∧ (𝑢𝑦) ∈ 𝑆)))
16 uneq2 4108 . . . . . 6 (𝑦 = 𝑣 → (𝑢𝑦) = (𝑢𝑣))
1716eleq1d 2898 . . . . 5 (𝑦 = 𝑣 → ((𝑢𝑦) ∈ 𝑆 ↔ (𝑢𝑣) ∈ 𝑆))
18 difeq2 4068 . . . . . 6 (𝑦 = 𝑣 → (𝑢𝑦) = (𝑢𝑣))
1918eleq1d 2898 . . . . 5 (𝑦 = 𝑣 → ((𝑢𝑦) ∈ 𝑆 ↔ (𝑢𝑣) ∈ 𝑆))
2017, 19anbi12d 633 . . . 4 (𝑦 = 𝑣 → (((𝑢𝑦) ∈ 𝑆 ∧ (𝑢𝑦) ∈ 𝑆) ↔ ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
2115, 20cbvral2vw 3436 . . 3 (∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆) ↔ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆))
22213anbi3i 1156 . 2 ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ (𝑥𝑦) ∈ 𝑆)) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
239, 10, 223bitr2i 302 1 (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  {crab 3134  cdif 3905  cun 3906  c0 4265  𝒫 cpw 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913
This theorem is referenced by:  rossspw  31502  0elros  31503  unelros  31504  difelros  31505
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