Theorem rossros 34637

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 34626	. . . 4 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂)
3		elpwg 4567	. . . 4 ⊢ (𝑆 ∈ 𝑄 → (𝑆 ∈ 𝒫 𝒫 𝑂 ↔ 𝑆 ⊆ 𝒫 𝑂))
4	2, 3	mpbird 260	. . 3 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝒫 𝒫 𝑂)
5	1	0elros 34627	. . 3 ⊢ (𝑆 ∈ 𝑄 → ∅ ∈ 𝑆)
6		uneq1 4115	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑢 ∪ 𝑣) = (𝑥 ∪ 𝑣))
7	6	eleq1d 2850	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → ((𝑢 ∪ 𝑣) ∈ 𝑠 ↔ (𝑥 ∪ 𝑣) ∈ 𝑠))
8		difeq1 4074	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑢 ∖ 𝑣) = (𝑥 ∖ 𝑣))
9	8	eleq1d 2850	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → ((𝑢 ∖ 𝑣) ∈ 𝑠 ↔ (𝑥 ∖ 𝑣) ∈ 𝑠))
10	7, 9	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑥 → (((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠) ↔ ((𝑥 ∪ 𝑣) ∈ 𝑠 ∧ (𝑥 ∖ 𝑣) ∈ 𝑠)))
11		uneq2 4116	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑥 ∪ 𝑣) = (𝑥 ∪ 𝑦))
12	11	eleq1d 2850	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → ((𝑥 ∪ 𝑣) ∈ 𝑠 ↔ (𝑥 ∪ 𝑦) ∈ 𝑠))
13		difeq2 4075	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑥 ∖ 𝑣) = (𝑥 ∖ 𝑦))
14	13	eleq1d 2850	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → ((𝑥 ∖ 𝑣) ∈ 𝑠 ↔ (𝑥 ∖ 𝑦) ∈ 𝑠))
15	12, 14	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑦 → (((𝑥 ∪ 𝑣) ∈ 𝑠 ∧ (𝑥 ∖ 𝑣) ∈ 𝑠) ↔ ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠)))
16	10, 15	cbvral2vw 3249	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 ((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠) ↔ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))
17	16	anbi2i 635	. . . . . . . . 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 3423	. . . . . . . 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 2791	. . . . . . 7 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 ((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠))}
20	19	inelros 34630	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∩ 𝑦) ∈ 𝑆)
21	20	3expb 1138	. . . . 5 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∩ 𝑦) ∈ 𝑆)
22	19	difelros 34629	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∖ 𝑦) ∈ 𝑆)
23	22	3expb 1138	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∖ 𝑦) ∈ 𝑆)
24	23	snssd 4754	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ⊆ 𝑆)
25		snex 5412	. . . . . . . 8 ⊢ {(𝑥 ∖ 𝑦)} ∈ V
26	25	elpw 4568	. . . . . . 7 ⊢ ({(𝑥 ∖ 𝑦)} ∈ 𝒫 𝑆 ↔ {(𝑥 ∖ 𝑦)} ⊆ 𝑆)
27	24, 26	sylibr 237	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ∈ 𝒫 𝑆)
28		snfi 9047	. . . . . . 7 ⊢ {(𝑥 ∖ 𝑦)} ∈ Fin
29	28	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ∈ Fin)
30		disjxsn 5105	. . . . . . 7 ⊢ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡
31	30	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡)
32		unisng 4892	. . . . . . . 8 ⊢ ((𝑥 ∖ 𝑦) ∈ 𝑆 → ∪ {(𝑥 ∖ 𝑦)} = (𝑥 ∖ 𝑦))
33	23, 32	syl 18	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∪ {(𝑥 ∖ 𝑦)} = (𝑥 ∖ 𝑦))
34	33	eqcomd 2771	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∖ 𝑦) = ∪ {(𝑥 ∖ 𝑦)})
35		eleq1 2853	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → (𝑧 ∈ Fin ↔ {(𝑥 ∖ 𝑦)} ∈ Fin))
36		disjeq1 5085	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → (Disj 𝑡 ∈ 𝑧 𝑡 ↔ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡))
37		unieq 4885	. . . . . . . . 9 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → ∪ 𝑧 = ∪ {(𝑥 ∖ 𝑦)})
38	37	eqeq2d 2776	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → ((𝑥 ∖ 𝑦) = ∪ 𝑧 ↔ (𝑥 ∖ 𝑦) = ∪ {(𝑥 ∖ 𝑦)}))
39	35, 36, 38	3anbi123d 1464	. . . . . . 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 3583	. . . . . 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 1399	. . . . 5 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧))
42	21, 41	jca 521	. . . 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 3210	. . 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 1146	. 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 34632	. 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 237	1 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝑁)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  {crab 3418   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  {csn 4591  ∪ cuni 4874  Disj wdisj 5078  Fincfn 8949

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-disj 5079  df-br 5112  df-opab 5176  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-om 7869  df-1o 8459  df-en 8950  df-fin 8953

This theorem is used by: (None)