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Theorem difelros 30209
Description: A ring of sets is closed under 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
difelros ((𝑆𝑄𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 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 difelros
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 1060 . . 3 ((𝑆𝑄𝐴𝑆𝐵𝑆) → 𝐴𝑆)
2 simp3 1061 . . 3 ((𝑆𝑄𝐴𝑆𝐵𝑆) → 𝐵𝑆)
3 isros.1 . . . . . 6 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
43isros 30205 . . . . 5 (𝑆𝑄 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)))
54simp3bi 1076 . . . 4 (𝑆𝑄 → ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆))
653ad2ant1 1080 . . 3 ((𝑆𝑄𝐴𝑆𝐵𝑆) → ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆))
7 uneq1 3752 . . . . . 6 (𝑢 = 𝐴 → (𝑢𝑣) = (𝐴𝑣))
87eleq1d 2684 . . . . 5 (𝑢 = 𝐴 → ((𝑢𝑣) ∈ 𝑆 ↔ (𝐴𝑣) ∈ 𝑆))
9 difeq1 3713 . . . . . 6 (𝑢 = 𝐴 → (𝑢𝑣) = (𝐴𝑣))
109eleq1d 2684 . . . . 5 (𝑢 = 𝐴 → ((𝑢𝑣) ∈ 𝑆 ↔ (𝐴𝑣) ∈ 𝑆))
118, 10anbi12d 746 . . . 4 (𝑢 = 𝐴 → (((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆) ↔ ((𝐴𝑣) ∈ 𝑆 ∧ (𝐴𝑣) ∈ 𝑆)))
12 uneq2 3753 . . . . . 6 (𝑣 = 𝐵 → (𝐴𝑣) = (𝐴𝐵))
1312eleq1d 2684 . . . . 5 (𝑣 = 𝐵 → ((𝐴𝑣) ∈ 𝑆 ↔ (𝐴𝐵) ∈ 𝑆))
14 difeq2 3714 . . . . . 6 (𝑣 = 𝐵 → (𝐴𝑣) = (𝐴𝐵))
1514eleq1d 2684 . . . . 5 (𝑣 = 𝐵 → ((𝐴𝑣) ∈ 𝑆 ↔ (𝐴𝐵) ∈ 𝑆))
1613, 15anbi12d 746 . . . 4 (𝑣 = 𝐵 → (((𝐴𝑣) ∈ 𝑆 ∧ (𝐴𝑣) ∈ 𝑆) ↔ ((𝐴𝐵) ∈ 𝑆 ∧ (𝐴𝐵) ∈ 𝑆)))
1711, 16rspc2va 3318 . . 3 (((𝐴𝑆𝐵𝑆) ∧ ∀𝑢𝑆𝑣𝑆 ((𝑢𝑣) ∈ 𝑆 ∧ (𝑢𝑣) ∈ 𝑆)) → ((𝐴𝐵) ∈ 𝑆 ∧ (𝐴𝐵) ∈ 𝑆))
181, 2, 6, 17syl21anc 1323 . 2 ((𝑆𝑄𝐴𝑆𝐵𝑆) → ((𝐴𝐵) ∈ 𝑆 ∧ (𝐴𝐵) ∈ 𝑆))
1918simprd 479 1 ((𝑆𝑄𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  {crab 2913  cdif 3564  cun 3565  c0 3907  𝒫 cpw 4149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572
This theorem is referenced by:  inelros  30210  rossros  30217
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