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Theorem rossros 32148
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 32137 . . . 4 (𝑆𝑄𝑆 ⊆ 𝒫 𝑂)
3 elpwg 4536 . . . 4 (𝑆𝑄 → (𝑆 ∈ 𝒫 𝒫 𝑂𝑆 ⊆ 𝒫 𝑂))
42, 3mpbird 256 . . 3 (𝑆𝑄𝑆 ∈ 𝒫 𝒫 𝑂)
510elros 32138 . . 3 (𝑆𝑄 → ∅ ∈ 𝑆)
6 uneq1 4090 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
76eleq1d 2823 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝑠 ↔ (𝑥𝑣) ∈ 𝑠))
8 difeq1 4050 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
98eleq1d 2823 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝑠 ↔ (𝑥𝑣) ∈ 𝑠))
107, 9anbi12d 631 . . . . . . . . . . 11 (𝑢 = 𝑥 → (((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠) ↔ ((𝑥𝑣) ∈ 𝑠 ∧ (𝑥𝑣) ∈ 𝑠)))
11 uneq2 4091 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
1211eleq1d 2823 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑠))
13 difeq2 4051 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
1413eleq1d 2823 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑠))
1512, 14anbi12d 631 . . . . . . . . . . 11 (𝑣 = 𝑦 → (((𝑥𝑣) ∈ 𝑠 ∧ (𝑥𝑣) ∈ 𝑠) ↔ ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠)))
1610, 15cbvral2vw 3396 . . . . . . . . . 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 3408 . . . . . . . 8 {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠))} = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
191, 18eqtr4i 2769 . . . . . . 7 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠))}
2019inelros 32141 . . . . . 6 ((𝑆𝑄𝑥𝑆𝑦𝑆) → (𝑥𝑦) ∈ 𝑆)
21203expb 1119 . . . . 5 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) ∈ 𝑆)
2219difelros 32140 . . . . . . . . 9 ((𝑆𝑄𝑥𝑆𝑦𝑆) → (𝑥𝑦) ∈ 𝑆)
23223expb 1119 . . . . . . . 8 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) ∈ 𝑆)
2423snssd 4742 . . . . . . 7 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ⊆ 𝑆)
25 snex 5354 . . . . . . . 8 {(𝑥𝑦)} ∈ V
2625elpw 4537 . . . . . . 7 ({(𝑥𝑦)} ∈ 𝒫 𝑆 ↔ {(𝑥𝑦)} ⊆ 𝑆)
2724, 26sylibr 233 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ∈ 𝒫 𝑆)
28 snfi 8834 . . . . . . 7 {(𝑥𝑦)} ∈ Fin
2928a1i 11 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ∈ Fin)
30 disjxsn 5067 . . . . . . 7 Disj 𝑡 ∈ {(𝑥𝑦)}𝑡
3130a1i 11 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → Disj 𝑡 ∈ {(𝑥𝑦)}𝑡)
32 unisng 4860 . . . . . . . 8 ((𝑥𝑦) ∈ 𝑆 {(𝑥𝑦)} = (𝑥𝑦))
3323, 32syl 17 . . . . . . 7 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} = (𝑥𝑦))
3433eqcomd 2744 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) = {(𝑥𝑦)})
35 eleq1 2826 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → (𝑧 ∈ Fin ↔ {(𝑥𝑦)} ∈ Fin))
36 disjeq1 5046 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → (Disj 𝑡𝑧 𝑡Disj 𝑡 ∈ {(𝑥𝑦)}𝑡))
37 unieq 4850 . . . . . . . . 9 (𝑧 = {(𝑥𝑦)} → 𝑧 = {(𝑥𝑦)})
3837eqeq2d 2749 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → ((𝑥𝑦) = 𝑧 ↔ (𝑥𝑦) = {(𝑥𝑦)}))
3935, 36, 383anbi123d 1435 . . . . . . 7 (𝑧 = {(𝑥𝑦)} → ((𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧) ↔ ({(𝑥𝑦)} ∈ Fin ∧ Disj 𝑡 ∈ {(𝑥𝑦)}𝑡 ∧ (𝑥𝑦) = {(𝑥𝑦)})))
4039rspcev 3561 . . . . . 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 1371 . . . . 5 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))
4221, 41jca 512 . . . 4 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
4342ralrimivva 3123 . . 3 (𝑆𝑄 → ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
444, 5, 433jca 1127 . 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 32143 . 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 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065  {crab 3068  cdif 3884  cun 3885  cin 3886  wss 3887  c0 4256  𝒫 cpw 4533  {csn 4561   cuni 4839  Disj wdisj 5039  Fincfn 8733
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-disj 5040  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-om 7713  df-1o 8297  df-en 8734  df-fin 8737
This theorem is referenced by: (None)
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