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Theorem issros 33665
Description: The property of being a semirings of sets, i.e., collections of sets containing the empty set, closed under finite intersection, and where complements can be written as finite disjoint unions. (Contributed by Thierry Arnoux, 18-Jul-2020.)
Hypothesis
Ref Expression
issros.1 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}
Assertion
Ref Expression
issros (𝑆𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦,𝑧
Allowed substitution hints:   𝑆(𝑡)   𝑁(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡)

Proof of Theorem issros
StepHypRef Expression
1 eleq2 2814 . . . 4 (𝑠 = 𝑆 → (∅ ∈ 𝑠 ↔ ∅ ∈ 𝑆))
2 eleq2 2814 . . . . . . 7 (𝑠 = 𝑆 → ((𝑥𝑦) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑆))
3 pweq 4609 . . . . . . . 8 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
43rexeqdv 3318 . . . . . . 7 (𝑠 = 𝑆 → (∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧) ↔ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
52, 4anbi12d 630 . . . . . 6 (𝑠 = 𝑆 → (((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)) ↔ ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
65raleqbi1dv 3325 . . . . 5 (𝑠 = 𝑆 → (∀𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)) ↔ ∀𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
76raleqbi1dv 3325 . . . 4 (𝑠 = 𝑆 → (∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)) ↔ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
81, 7anbi12d 630 . . 3 (𝑠 = 𝑆 → ((∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))) ↔ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))))
9 issros.1 . . 3 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}
108, 9elrab2 3679 . 2 (𝑆𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))))
11 3anass 1092 . 2 ((𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))) ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))))
1210, 11bitr4i 278 1 (𝑆𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  wrex 3062  {crab 3424  cdif 3938  cin 3940  c0 4315  𝒫 cpw 4595   cuni 4900  Disj wdisj 5104  Fincfn 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-in 3948  df-ss 3958  df-pw 4597
This theorem is referenced by:  srossspw  33666  0elsros  33667  inelsros  33668  diffiunisros  33669  rossros  33670
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