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Theorem rossros 30759
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 30748 . . . 4 (𝑆𝑄𝑆 ⊆ 𝒫 𝑂)
3 elpwg 4357 . . . 4 (𝑆𝑄 → (𝑆 ∈ 𝒫 𝒫 𝑂𝑆 ⊆ 𝒫 𝑂))
42, 3mpbird 249 . . 3 (𝑆𝑄𝑆 ∈ 𝒫 𝒫 𝑂)
510elros 30749 . . 3 (𝑆𝑄 → ∅ ∈ 𝑆)
6 uneq1 3958 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
76eleq1d 2863 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝑠 ↔ (𝑥𝑣) ∈ 𝑠))
8 difeq1 3919 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
98eleq1d 2863 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝑠 ↔ (𝑥𝑣) ∈ 𝑠))
107, 9anbi12d 625 . . . . . . . . . . 11 (𝑢 = 𝑥 → (((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠) ↔ ((𝑥𝑣) ∈ 𝑠 ∧ (𝑥𝑣) ∈ 𝑠)))
11 uneq2 3959 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
1211eleq1d 2863 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑠))
13 difeq2 3920 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
1413eleq1d 2863 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑠))
1512, 14anbi12d 625 . . . . . . . . . . 11 (𝑣 = 𝑦 → (((𝑥𝑣) ∈ 𝑠 ∧ (𝑥𝑣) ∈ 𝑠) ↔ ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠)))
1610, 15cbvral2v 3362 . . . . . . . . . 10 (∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠) ↔ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))
1716anbi2i 617 . . . . . . . . 9 ((∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠)) ↔ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠)))
1817rabbii 3369 . . . . . . . 8 {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠))} = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
191, 18eqtr4i 2824 . . . . . . 7 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠))}
2019inelros 30752 . . . . . 6 ((𝑆𝑄𝑥𝑆𝑦𝑆) → (𝑥𝑦) ∈ 𝑆)
21203expb 1150 . . . . 5 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) ∈ 𝑆)
2219difelros 30751 . . . . . . . . 9 ((𝑆𝑄𝑥𝑆𝑦𝑆) → (𝑥𝑦) ∈ 𝑆)
23223expb 1150 . . . . . . . 8 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) ∈ 𝑆)
2423snssd 4528 . . . . . . 7 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ⊆ 𝑆)
25 snex 5099 . . . . . . . 8 {(𝑥𝑦)} ∈ V
2625elpw 4355 . . . . . . 7 ({(𝑥𝑦)} ∈ 𝒫 𝑆 ↔ {(𝑥𝑦)} ⊆ 𝑆)
2724, 26sylibr 226 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ∈ 𝒫 𝑆)
28 snfi 8280 . . . . . . 7 {(𝑥𝑦)} ∈ Fin
2928a1i 11 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ∈ Fin)
30 disjxsn 4837 . . . . . . 7 Disj 𝑡 ∈ {(𝑥𝑦)}𝑡
3130a1i 11 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → Disj 𝑡 ∈ {(𝑥𝑦)}𝑡)
32 unisng 4643 . . . . . . . 8 ((𝑥𝑦) ∈ 𝑆 {(𝑥𝑦)} = (𝑥𝑦))
3323, 32syl 17 . . . . . . 7 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} = (𝑥𝑦))
3433eqcomd 2805 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) = {(𝑥𝑦)})
35 eleq1 2866 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → (𝑧 ∈ Fin ↔ {(𝑥𝑦)} ∈ Fin))
36 disjeq1 4818 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → (Disj 𝑡𝑧 𝑡Disj 𝑡 ∈ {(𝑥𝑦)}𝑡))
37 unieq 4636 . . . . . . . . 9 (𝑧 = {(𝑥𝑦)} → 𝑧 = {(𝑥𝑦)})
3837eqeq2d 2809 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → ((𝑥𝑦) = 𝑧 ↔ (𝑥𝑦) = {(𝑥𝑦)}))
3935, 36, 383anbi123d 1561 . . . . . . 7 (𝑧 = {(𝑥𝑦)} → ((𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧) ↔ ({(𝑥𝑦)} ∈ Fin ∧ Disj 𝑡 ∈ {(𝑥𝑦)}𝑡 ∧ (𝑥𝑦) = {(𝑥𝑦)})))
4039rspcev 3497 . . . . . 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 1492 . . . . 5 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))
4221, 41jca 508 . . . 4 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
4342ralrimivva 3152 . . 3 (𝑆𝑄 → ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
444, 5, 433jca 1159 . 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 30754 . 2 (𝑆𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
4744, 46sylibr 226 1 (𝑆𝑄𝑆𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089  wrex 3090  {crab 3093  cdif 3766  cun 3767  cin 3768  wss 3769  c0 4115  𝒫 cpw 4349  {csn 4368   cuni 4628  Disj wdisj 4811  Fincfn 8195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-disj 4812  df-br 4844  df-opab 4906  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-om 7300  df-1o 7799  df-en 8196  df-fin 8199
This theorem is referenced by: (None)
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