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Theorem rossros 33209
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 33198 . . . 4 (𝑆𝑄𝑆 ⊆ 𝒫 𝑂)
3 elpwg 4606 . . . 4 (𝑆𝑄 → (𝑆 ∈ 𝒫 𝒫 𝑂𝑆 ⊆ 𝒫 𝑂))
42, 3mpbird 257 . . 3 (𝑆𝑄𝑆 ∈ 𝒫 𝒫 𝑂)
510elros 33199 . . 3 (𝑆𝑄 → ∅ ∈ 𝑆)
6 uneq1 4157 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
76eleq1d 2819 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝑠 ↔ (𝑥𝑣) ∈ 𝑠))
8 difeq1 4116 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
98eleq1d 2819 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝑠 ↔ (𝑥𝑣) ∈ 𝑠))
107, 9anbi12d 632 . . . . . . . . . . 11 (𝑢 = 𝑥 → (((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠) ↔ ((𝑥𝑣) ∈ 𝑠 ∧ (𝑥𝑣) ∈ 𝑠)))
11 uneq2 4158 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
1211eleq1d 2819 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑠))
13 difeq2 4117 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
1413eleq1d 2819 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑠))
1512, 14anbi12d 632 . . . . . . . . . . 11 (𝑣 = 𝑦 → (((𝑥𝑣) ∈ 𝑠 ∧ (𝑥𝑣) ∈ 𝑠) ↔ ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠)))
1610, 15cbvral2vw 3239 . . . . . . . . . 10 (∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠) ↔ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))
1716anbi2i 624 . . . . . . . . 9 ((∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠)) ↔ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠)))
1817rabbii 3439 . . . . . . . 8 {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠))} = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
191, 18eqtr4i 2764 . . . . . . 7 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠))}
2019inelros 33202 . . . . . 6 ((𝑆𝑄𝑥𝑆𝑦𝑆) → (𝑥𝑦) ∈ 𝑆)
21203expb 1121 . . . . 5 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) ∈ 𝑆)
2219difelros 33201 . . . . . . . . 9 ((𝑆𝑄𝑥𝑆𝑦𝑆) → (𝑥𝑦) ∈ 𝑆)
23223expb 1121 . . . . . . . 8 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) ∈ 𝑆)
2423snssd 4813 . . . . . . 7 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ⊆ 𝑆)
25 snex 5432 . . . . . . . 8 {(𝑥𝑦)} ∈ V
2625elpw 4607 . . . . . . 7 ({(𝑥𝑦)} ∈ 𝒫 𝑆 ↔ {(𝑥𝑦)} ⊆ 𝑆)
2724, 26sylibr 233 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ∈ 𝒫 𝑆)
28 snfi 9044 . . . . . . 7 {(𝑥𝑦)} ∈ Fin
2928a1i 11 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ∈ Fin)
30 disjxsn 5142 . . . . . . 7 Disj 𝑡 ∈ {(𝑥𝑦)}𝑡
3130a1i 11 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → Disj 𝑡 ∈ {(𝑥𝑦)}𝑡)
32 unisng 4930 . . . . . . . 8 ((𝑥𝑦) ∈ 𝑆 {(𝑥𝑦)} = (𝑥𝑦))
3323, 32syl 17 . . . . . . 7 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} = (𝑥𝑦))
3433eqcomd 2739 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) = {(𝑥𝑦)})
35 eleq1 2822 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → (𝑧 ∈ Fin ↔ {(𝑥𝑦)} ∈ Fin))
36 disjeq1 5121 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → (Disj 𝑡𝑧 𝑡Disj 𝑡 ∈ {(𝑥𝑦)}𝑡))
37 unieq 4920 . . . . . . . . 9 (𝑧 = {(𝑥𝑦)} → 𝑧 = {(𝑥𝑦)})
3837eqeq2d 2744 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → ((𝑥𝑦) = 𝑧 ↔ (𝑥𝑦) = {(𝑥𝑦)}))
3935, 36, 383anbi123d 1437 . . . . . . 7 (𝑧 = {(𝑥𝑦)} → ((𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧) ↔ ({(𝑥𝑦)} ∈ Fin ∧ Disj 𝑡 ∈ {(𝑥𝑦)}𝑡 ∧ (𝑥𝑦) = {(𝑥𝑦)})))
4039rspcev 3613 . . . . . 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 1373 . . . . 5 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))
4221, 41jca 513 . . . 4 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
4342ralrimivva 3201 . . 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 33204 . 2 (𝑆𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
4744, 46sylibr 233 1 (𝑆𝑄𝑆𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  wrex 3071  {crab 3433  cdif 3946  cun 3947  cin 3948  wss 3949  c0 4323  𝒫 cpw 4603  {csn 4629   cuni 4909  Disj wdisj 5114  Fincfn 8939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-disj 5115  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-om 7856  df-1o 8466  df-en 8940  df-fin 8943
This theorem is referenced by: (None)
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