Theorem rossros 32868

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 32857	. . . 4 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂)
3		elpwg 4568	. . . 4 ⊢ (𝑆 ∈ 𝑄 → (𝑆 ∈ 𝒫 𝒫 𝑂 ↔ 𝑆 ⊆ 𝒫 𝑂))
4	2, 3	mpbird 256	. . 3 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝒫 𝒫 𝑂)
5	1	0elros 32858	. . 3 ⊢ (𝑆 ∈ 𝑄 → ∅ ∈ 𝑆)
6		uneq1 4121	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑢 ∪ 𝑣) = (𝑥 ∪ 𝑣))
7	6	eleq1d 2817	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → ((𝑢 ∪ 𝑣) ∈ 𝑠 ↔ (𝑥 ∪ 𝑣) ∈ 𝑠))
8		difeq1 4080	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑢 ∖ 𝑣) = (𝑥 ∖ 𝑣))
9	8	eleq1d 2817	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → ((𝑢 ∖ 𝑣) ∈ 𝑠 ↔ (𝑥 ∖ 𝑣) ∈ 𝑠))
10	7, 9	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑥 → (((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠) ↔ ((𝑥 ∪ 𝑣) ∈ 𝑠 ∧ (𝑥 ∖ 𝑣) ∈ 𝑠)))
11		uneq2 4122	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑥 ∪ 𝑣) = (𝑥 ∪ 𝑦))
12	11	eleq1d 2817	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → ((𝑥 ∪ 𝑣) ∈ 𝑠 ↔ (𝑥 ∪ 𝑦) ∈ 𝑠))
13		difeq2 4081	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑥 ∖ 𝑣) = (𝑥 ∖ 𝑦))
14	13	eleq1d 2817	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → ((𝑥 ∖ 𝑣) ∈ 𝑠 ↔ (𝑥 ∖ 𝑦) ∈ 𝑠))
15	12, 14	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑦 → (((𝑥 ∪ 𝑣) ∈ 𝑠 ∧ (𝑥 ∖ 𝑣) ∈ 𝑠) ↔ ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠)))
16	10, 15	cbvral2vw 3225	. . . . . . . . . 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 3411	. . . . . . . 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 2762	. . . . . . 7 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 ((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠))}
20	19	inelros 32861	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∩ 𝑦) ∈ 𝑆)
21	20	3expb 1120	. . . . 5 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∩ 𝑦) ∈ 𝑆)
22	19	difelros 32860	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∖ 𝑦) ∈ 𝑆)
23	22	3expb 1120	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∖ 𝑦) ∈ 𝑆)
24	23	snssd 4774	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ⊆ 𝑆)
25		snex 5393	. . . . . . . 8 ⊢ {(𝑥 ∖ 𝑦)} ∈ V
26	25	elpw 4569	. . . . . . 7 ⊢ ({(𝑥 ∖ 𝑦)} ∈ 𝒫 𝑆 ↔ {(𝑥 ∖ 𝑦)} ⊆ 𝑆)
27	24, 26	sylibr 233	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ∈ 𝒫 𝑆)
28		snfi 8995	. . . . . . 7 ⊢ {(𝑥 ∖ 𝑦)} ∈ Fin
29	28	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ∈ Fin)
30		disjxsn 5103	. . . . . . 7 ⊢ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡
31	30	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡)
32		unisng 4891	. . . . . . . 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 2737	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∖ 𝑦) = ∪ {(𝑥 ∖ 𝑦)})
35		eleq1 2820	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → (𝑧 ∈ Fin ↔ {(𝑥 ∖ 𝑦)} ∈ Fin))
36		disjeq1 5082	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → (Disj 𝑡 ∈ 𝑧 𝑡 ↔ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡))
37		unieq 4881	. . . . . . . . 9 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → ∪ 𝑧 = ∪ {(𝑥 ∖ 𝑦)})
38	37	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → ((𝑥 ∖ 𝑦) = ∪ 𝑧 ↔ (𝑥 ∖ 𝑦) = ∪ {(𝑥 ∖ 𝑦)}))
39	35, 36, 38	3anbi123d 1436	. . . . . . 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 3582	. . . . . 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 1372	. . . . 5 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧))
42	21, 41	jca 512	. . . 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 3193	. . 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 1128	. 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 32863	. 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 233	1 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝑁)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069  {crab 3405   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4287  𝒫 cpw 4565  {csn 4591  ∪ cuni 4870  Disj wdisj 5075  Fincfn 8890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3351  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-disj 5076  df-br 5111  df-opab 5173  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-om 7808  df-1o 8417  df-en 8891  df-fin 8894

This theorem is referenced by: (None)