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Theorem rossros 30582
Description: Rings of sets are semi-rings 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 30571 . . . 4 (𝑆𝑄𝑆 ⊆ 𝒫 𝑂)
3 elpwg 4306 . . . 4 (𝑆𝑄 → (𝑆 ∈ 𝒫 𝒫 𝑂𝑆 ⊆ 𝒫 𝑂))
42, 3mpbird 247 . . 3 (𝑆𝑄𝑆 ∈ 𝒫 𝒫 𝑂)
510elros 30572 . . 3 (𝑆𝑄 → ∅ ∈ 𝑆)
6 uneq1 3911 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
76eleq1d 2835 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝑠 ↔ (𝑥𝑣) ∈ 𝑠))
8 difeq1 3872 . . . . . . . . . . . . 13 (𝑢 = 𝑥 → (𝑢𝑣) = (𝑥𝑣))
98eleq1d 2835 . . . . . . . . . . . 12 (𝑢 = 𝑥 → ((𝑢𝑣) ∈ 𝑠 ↔ (𝑥𝑣) ∈ 𝑠))
107, 9anbi12d 616 . . . . . . . . . . 11 (𝑢 = 𝑥 → (((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠) ↔ ((𝑥𝑣) ∈ 𝑠 ∧ (𝑥𝑣) ∈ 𝑠)))
11 uneq2 3912 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
1211eleq1d 2835 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑠))
13 difeq2 3873 . . . . . . . . . . . . 13 (𝑣 = 𝑦 → (𝑥𝑣) = (𝑥𝑦))
1413eleq1d 2835 . . . . . . . . . . . 12 (𝑣 = 𝑦 → ((𝑥𝑣) ∈ 𝑠 ↔ (𝑥𝑦) ∈ 𝑠))
1512, 14anbi12d 616 . . . . . . . . . . 11 (𝑣 = 𝑦 → (((𝑥𝑣) ∈ 𝑠 ∧ (𝑥𝑣) ∈ 𝑠) ↔ ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠)))
1610, 15cbvral2v 3328 . . . . . . . . . 10 (∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠) ↔ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))
1716anbi2i 609 . . . . . . . . 9 ((∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠)) ↔ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠)))
1817rabbii 3335 . . . . . . . 8 {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠))} = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠𝑦𝑠 ((𝑥𝑦) ∈ 𝑠 ∧ (𝑥𝑦) ∈ 𝑠))}
191, 18eqtr4i 2796 . . . . . . 7 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢𝑠𝑣𝑠 ((𝑢𝑣) ∈ 𝑠 ∧ (𝑢𝑣) ∈ 𝑠))}
2019inelros 30575 . . . . . 6 ((𝑆𝑄𝑥𝑆𝑦𝑆) → (𝑥𝑦) ∈ 𝑆)
21203expb 1113 . . . . 5 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) ∈ 𝑆)
2219difelros 30574 . . . . . . . . 9 ((𝑆𝑄𝑥𝑆𝑦𝑆) → (𝑥𝑦) ∈ 𝑆)
23223expb 1113 . . . . . . . 8 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) ∈ 𝑆)
2423snssd 4476 . . . . . . 7 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ⊆ 𝑆)
25 snex 5037 . . . . . . . 8 {(𝑥𝑦)} ∈ V
2625elpw 4304 . . . . . . 7 ({(𝑥𝑦)} ∈ 𝒫 𝑆 ↔ {(𝑥𝑦)} ⊆ 𝑆)
2724, 26sylibr 224 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ∈ 𝒫 𝑆)
28 snfi 8197 . . . . . . 7 {(𝑥𝑦)} ∈ Fin
2928a1i 11 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} ∈ Fin)
30 disjxsn 4781 . . . . . . 7 Disj 𝑡 ∈ {(𝑥𝑦)}𝑡
3130a1i 11 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → Disj 𝑡 ∈ {(𝑥𝑦)}𝑡)
32 unisng 4591 . . . . . . . 8 ((𝑥𝑦) ∈ 𝑆 {(𝑥𝑦)} = (𝑥𝑦))
3323, 32syl 17 . . . . . . 7 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → {(𝑥𝑦)} = (𝑥𝑦))
3433eqcomd 2777 . . . . . 6 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝑦) = {(𝑥𝑦)})
35 eleq1 2838 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → (𝑧 ∈ Fin ↔ {(𝑥𝑦)} ∈ Fin))
36 disjeq1 4762 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → (Disj 𝑡𝑧 𝑡Disj 𝑡 ∈ {(𝑥𝑦)}𝑡))
37 unieq 4583 . . . . . . . . 9 (𝑧 = {(𝑥𝑦)} → 𝑧 = {(𝑥𝑦)})
3837eqeq2d 2781 . . . . . . . 8 (𝑧 = {(𝑥𝑦)} → ((𝑥𝑦) = 𝑧 ↔ (𝑥𝑦) = {(𝑥𝑦)}))
3935, 36, 383anbi123d 1547 . . . . . . 7 (𝑧 = {(𝑥𝑦)} → ((𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧) ↔ ({(𝑥𝑦)} ∈ Fin ∧ Disj 𝑡 ∈ {(𝑥𝑦)}𝑡 ∧ (𝑥𝑦) = {(𝑥𝑦)})))
4039rspcev 3460 . . . . . 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 1478 . . . . 5 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))
4221, 41jca 501 . . . 4 ((𝑆𝑄 ∧ (𝑥𝑆𝑦𝑆)) → ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
4342ralrimivva 3120 . . 3 (𝑆𝑄 → ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧)))
444, 5, 433jca 1122 . 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 30577 . 2 (𝑆𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 ((𝑥𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡𝑧 𝑡 ∧ (𝑥𝑦) = 𝑧))))
4744, 46sylibr 224 1 (𝑆𝑄𝑆𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  {crab 3065  cdif 3720  cun 3721  cin 3722  wss 3723  c0 4063  𝒫 cpw 4298  {csn 4317   cuni 4575  Disj wdisj 4755  Fincfn 8112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-disj 4756  df-br 4788  df-opab 4848  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-om 7216  df-1o 7716  df-en 8113  df-fin 8116
This theorem is referenced by: (None)
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