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Theorem diffiunisros 33472
Description: In semiring of sets, 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
diffiunisros ((𝑆𝑁𝐴𝑆𝐵𝑆) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧))
Distinct variable groups:   𝑡,𝑠,𝑥,𝑦   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧
Allowed substitution hints:   𝐴(𝑡,𝑠)   𝐵(𝑥,𝑡,𝑠)   𝑆(𝑡)   𝑁(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡)

Proof of Theorem diffiunisros
StepHypRef Expression
1 simp2 1136 . . 3 ((𝑆𝑁𝐴𝑆𝐵𝑆) → 𝐴𝑆)
2 simp3 1137 . . 3 ((𝑆𝑁𝐴𝑆𝐵𝑆) → 𝐵𝑆)
3 issros.1 . . . . . 6 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}
43issros 33468 . . . . 5 (𝑆𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
54simp3bi 1146 . . . 4 (𝑆𝑁 → ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
653ad2ant1 1132 . . 3 ((𝑆𝑁𝐴𝑆𝐵𝑆) → ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
7 ineq1 4206 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
87eleq1d 2817 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ 𝑆 ↔ (𝐴𝑦) ∈ 𝑆))
9 difeq1 4116 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
109eqeq1d 2733 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝑦) = 𝑧 ↔ (𝐴𝑦) = 𝑧))
11103anbi3d 1441 . . . . . 6 (𝑥 = 𝐴 → ((𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧) ↔ (𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧)))
1211rexbidv 3177 . . . . 5 (𝑥 = 𝐴 → (∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧) ↔ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧)))
138, 12anbi12d 630 . . . 4 (𝑥 = 𝐴 → (((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)) ↔ ((𝐴𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧))))
14 ineq2 4207 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
1514eleq1d 2817 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ 𝑆 ↔ (𝐴𝐵) ∈ 𝑆))
16 difeq2 4117 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
1716eqeq1d 2733 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴𝑦) = 𝑧 ↔ (𝐴𝐵) = 𝑧))
18173anbi3d 1441 . . . . . 6 (𝑦 = 𝐵 → ((𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧) ↔ (𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧)))
1918rexbidv 3177 . . . . 5 (𝑦 = 𝐵 → (∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧) ↔ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧)))
2015, 19anbi12d 630 . . . 4 (𝑦 = 𝐵 → (((𝐴𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝑦) = 𝑧)) ↔ ((𝐴𝐵) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧))))
2113, 20rspc2va 3624 . . 3 (((𝐴𝑆𝐵𝑆) ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))) → ((𝐴𝐵) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧)))
221, 2, 6, 21syl21anc 835 . 2 ((𝑆𝑁𝐴𝑆𝐵𝑆) → ((𝐴𝐵) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧)))
2322simprd 495 1 ((𝑆𝑁𝐴𝑆𝐵𝑆) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝐴𝐵) = 𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wrex 3069  {crab 3431  cdif 3946  cin 3948  c0 4323  𝒫 cpw 4603   cuni 4909  Disj wdisj 5114  Fincfn 8942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-in 3956  df-ss 3966  df-pw 4605
This theorem is referenced by: (None)
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