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 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}
rossros.n	⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}

Assertion

Ref	Expression
rossros	⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝑁)

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.

Step	Hyp	Ref	Expression
1		rossros.q	. . . . 5 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}
2	1	rossspw 34170	. . . 4 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂)
3		elpwg 4603	. . . 4 ⊢ (𝑆 ∈ 𝑄 → (𝑆 ∈ 𝒫 𝒫 𝑂 ↔ 𝑆 ⊆ 𝒫 𝑂))
4	2, 3	mpbird 257	. . 3 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝒫 𝒫 𝑂)
5	1	0elros 34171	. . 3 ⊢ (𝑆 ∈ 𝑄 → ∅ ∈ 𝑆)
6		uneq1 4161	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑢 ∪ 𝑣) = (𝑥 ∪ 𝑣))
7	6	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → ((𝑢 ∪ 𝑣) ∈ 𝑠 ↔ (𝑥 ∪ 𝑣) ∈ 𝑠))
8		difeq1 4119	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑢 ∖ 𝑣) = (𝑥 ∖ 𝑣))
9	8	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → ((𝑢 ∖ 𝑣) ∈ 𝑠 ↔ (𝑥 ∖ 𝑣) ∈ 𝑠))
10	7, 9	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑥 → (((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠) ↔ ((𝑥 ∪ 𝑣) ∈ 𝑠 ∧ (𝑥 ∖ 𝑣) ∈ 𝑠)))
11		uneq2 4162	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑥 ∪ 𝑣) = (𝑥 ∪ 𝑦))
12	11	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → ((𝑥 ∪ 𝑣) ∈ 𝑠 ↔ (𝑥 ∪ 𝑦) ∈ 𝑠))
13		difeq2 4120	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑥 ∖ 𝑣) = (𝑥 ∖ 𝑦))
14	13	eleq1d 2826	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → ((𝑥 ∖ 𝑣) ∈ 𝑠 ↔ (𝑥 ∖ 𝑦) ∈ 𝑠))
15	12, 14	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑦 → (((𝑥 ∪ 𝑣) ∈ 𝑠 ∧ (𝑥 ∖ 𝑣) ∈ 𝑠) ↔ ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠)))
16	10, 15	cbvral2vw 3241	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 ((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠) ↔ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))
17	16	anbi2i 623	. . . . . . . . 9 ⊢ ((∅ ∈ 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 ((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠)) ↔ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠)))
18	17	rabbii 3442	. . . . . . . 8 ⊢ {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 ((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠))} = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))}
19	1, 18	eqtr4i 2768	. . . . . . 7 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 ((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠))}
20	19	inelros 34174	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∩ 𝑦) ∈ 𝑆)
21	20	3expb 1121	. . . . 5 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∩ 𝑦) ∈ 𝑆)
22	19	difelros 34173	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∖ 𝑦) ∈ 𝑆)
23	22	3expb 1121	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∖ 𝑦) ∈ 𝑆)
24	23	snssd 4809	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ⊆ 𝑆)
25		snex 5436	. . . . . . . 8 ⊢ {(𝑥 ∖ 𝑦)} ∈ V
26	25	elpw 4604	. . . . . . 7 ⊢ ({(𝑥 ∖ 𝑦)} ∈ 𝒫 𝑆 ↔ {(𝑥 ∖ 𝑦)} ⊆ 𝑆)
27	24, 26	sylibr 234	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ∈ 𝒫 𝑆)
28		snfi 9083	. . . . . . 7 ⊢ {(𝑥 ∖ 𝑦)} ∈ Fin
29	28	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ∈ Fin)
30		disjxsn 5137	. . . . . . 7 ⊢ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡
31	30	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡)
32		unisng 4925	. . . . . . . 8 ⊢ ((𝑥 ∖ 𝑦) ∈ 𝑆 → ∪ {(𝑥 ∖ 𝑦)} = (𝑥 ∖ 𝑦))
33	23, 32	syl 17	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∪ {(𝑥 ∖ 𝑦)} = (𝑥 ∖ 𝑦))
34	33	eqcomd 2743	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∖ 𝑦) = ∪ {(𝑥 ∖ 𝑦)})
35		eleq1 2829	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → (𝑧 ∈ Fin ↔ {(𝑥 ∖ 𝑦)} ∈ Fin))
36		disjeq1 5117	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → (Disj 𝑡 ∈ 𝑧 𝑡 ↔ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡))
37		unieq 4918	. . . . . . . . 9 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → ∪ 𝑧 = ∪ {(𝑥 ∖ 𝑦)})
38	37	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → ((𝑥 ∖ 𝑦) = ∪ 𝑧 ↔ (𝑥 ∖ 𝑦) = ∪ {(𝑥 ∖ 𝑦)}))
39	35, 36, 38	3anbi123d 1438	. . . . . . 7 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → ((𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧) ↔ ({(𝑥 ∖ 𝑦)} ∈ Fin ∧ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ {(𝑥 ∖ 𝑦)})))
40	39	rspcev 3622	. . . . . 6 ⊢ (({(𝑥 ∖ 𝑦)} ∈ 𝒫 𝑆 ∧ ({(𝑥 ∖ 𝑦)} ∈ Fin ∧ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ {(𝑥 ∖ 𝑦)})) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧))
41	27, 29, 31, 34, 40	syl13anc 1374	. . . . 5 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧))
42	21, 41	jca 511	. . . 4 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ((𝑥 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))
43	42	ralrimivva 3202	. . 3 ⊢ (𝑆 ∈ 𝑄 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))
44	4, 5, 43	3jca 1129	. 2 ⊢ (𝑆 ∈ 𝑄 → (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧))))
45		rossros.n	. . 3 ⊢ 𝑁 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∩ 𝑦) ∈ 𝑠 ∧ ∃𝑧 ∈ 𝒫 𝑠(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))}
46	45	issros 34176	. 2 ⊢ (𝑆 ∈ 𝑁 ↔ (𝑆 ∈ 𝒫 𝒫 𝑂 ∧ ∅ ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧))))
47	44, 46	sylibr 234	1 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝑁)

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)