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Theorem rossros 34181
Description: Rings of sets are semirings of sets. (Contributed by Thierry Arnoux, 18-Jul-2020.)
Hypotheses
Ref Expression
rossros.q 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
rossros.n 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}
Assertion
Ref Expression
rossros (𝑆𝑄𝑆𝑁)
Distinct variable groups:   𝑂,𝑠   𝑥,𝑄,𝑦   𝑆,𝑠,𝑥,𝑦,𝑧   𝑡,𝑠,𝑥,𝑦,𝑧
Allowed substitution hints:   𝑄(𝑧,𝑡,𝑠)   𝑆(𝑡)   𝑁(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡)

Proof of Theorem rossros
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rossros.q . . . . 5 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
21rossspw 34170 . . . 4 (𝑆𝑄𝑆 ⊆ 𝒫 𝑂)
3 elpwg 4603 . . . 4 (𝑆𝑄 → (𝑆 ∈ 𝒫 𝒫 𝑂𝑆 ⊆ 𝒫 𝑂))
42, 3mpbird 257 . . 3 (𝑆𝑄𝑆 ∈ 𝒫 𝒫 𝑂)
510elros 34171 . . 3 (𝑆𝑄 → ∅ ∈ 𝑆)
6 uneq1 4161 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
76eleq1d 2826 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝑠 ↔ (𝑥𝑣) ∈ 𝑠))
8 difeq1 4119 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
98eleq1d 2826 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝑠 ↔ (𝑥𝑣) ∈ 𝑠))
107, 9anbi12d 632 . . . . . . . . . . 11 (𝑢 = 𝑥 → (((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠) ↔ ((𝑥𝑣) ∈ 𝑠 ∧ (𝑥𝑣) ∈ 𝑠)))
11 uneq2 4162 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
1211eleq1d 2826 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑠))
13 difeq2 4120 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
1413eleq1d 2826 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑠))
1512, 14anbi12d 632 . . . . . . . . . . 11 (𝑣 = 𝑦 → (((𝑥𝑣) ∈ 𝑠 ∧ (𝑥𝑣) ∈ 𝑠) ↔ ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠)))
1610, 15cbvral2vw 3241 . . . . . . . . . 10 (∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠) ↔ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))
1716anbi2i 623 . . . . . . . . 9 ((∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠)) ↔ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠)))
1817rabbii 3442 . . . . . . . 8 {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠))} = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
191, 18eqtr4i 2768 . . . . . . 7 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠))}
2019inelros 34174 . . . . . 6 ((𝑆𝑄𝑥𝑆𝑦𝑆) → (𝑥𝑦) ∈ 𝑆)
21203expb 1121 . . . . 5 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) ∈ 𝑆)
2219difelros 34173 . . . . . . . . 9 ((𝑆𝑄𝑥𝑆𝑦𝑆) → (𝑥𝑦) ∈ 𝑆)
23223expb 1121 . . . . . . . 8 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) ∈ 𝑆)
2423snssd 4809 . . . . . . 7 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ⊆ 𝑆)
25 snex 5436 . . . . . . . 8 {(𝑥𝑦)} ∈ V
2625elpw 4604 . . . . . . 7 ({(𝑥𝑦)} ∈ 𝒫 𝑆 ↔ {(𝑥𝑦)} ⊆ 𝑆)
2724, 26sylibr 234 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ∈ 𝒫 𝑆)
28 snfi 9083 . . . . . . 7 {(𝑥𝑦)} ∈ Fin
2928a1i 11 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ∈ Fin)
30 disjxsn 5137 . . . . . . 7 Disj 𝑡 ∈ {(𝑥𝑦)}𝑡
3130a1i 11 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → Disj 𝑡 ∈ {(𝑥𝑦)}𝑡)
32 unisng 4925 . . . . . . . 8 ((𝑥𝑦) ∈ 𝑆 {(𝑥𝑦)} = (𝑥𝑦))
3323, 32syl 17 . . . . . . 7 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} = (𝑥𝑦))
3433eqcomd 2743 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) = {(𝑥𝑦)})
35 eleq1 2829 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → (𝑧 ∈ Fin ↔ {(𝑥𝑦)} ∈ Fin))
36 disjeq1 5117 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → (Disj 𝑡𝑧 𝑡Disj 𝑡 ∈ {(𝑥𝑦)}𝑡))
37 unieq 4918 . . . . . . . . 9 (𝑧 = {(𝑥𝑦)} → 𝑧 = {(𝑥𝑦)})
3837eqeq2d 2748 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → ((𝑥𝑦) = 𝑧 ↔ (𝑥𝑦) = {(𝑥𝑦)}))
3935, 36, 383anbi123d 1438 . . . . . . 7 (𝑧 = {(𝑥𝑦)} → ((𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧) ↔ ({(𝑥𝑦)} ∈ Fin ∧ Disj 𝑡 ∈ {(𝑥𝑦)}𝑡 ∧ (𝑥𝑦) = {(𝑥𝑦)})))
4039rspcev 3622 . . . . . 6 (({(𝑥𝑦)} ∈ 𝒫 𝑆 ∧ ({(𝑥𝑦)} ∈ Fin ∧ Disj 𝑡 ∈ {(𝑥𝑦)}𝑡 ∧ (𝑥𝑦) = {(𝑥𝑦)})) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))
4127, 29, 31, 34, 40syl13anc 1374 . . . . 5 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))
4221, 41jca 511 . . . 4 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
4342ralrimivva 3202 . . 3 (𝑆𝑄 → ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
444, 5, 433jca 1129 . 2 (𝑆𝑄 → (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
45 rossros.n . . 3 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))}
4645issros 34176 . 2 (𝑆𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
4744, 46sylibr 234 1 (𝑆𝑄𝑆𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   cuni 4907  Disj wdisj 5110  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-disj 5111  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-om 7888  df-1o 8506  df-en 8986  df-fin 8989
This theorem is referenced by: (None)
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