Theorem rossros 34286

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 34275	. . . 4 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂)
3		elpwg 4555	. . . 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 34276	. . 3 ⊢ (𝑆 ∈ 𝑄 → ∅ ∈ 𝑆)
6		uneq1 4111	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑢 ∪ 𝑣) = (𝑥 ∪ 𝑣))
7	6	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → ((𝑢 ∪ 𝑣) ∈ 𝑠 ↔ (𝑥 ∪ 𝑣) ∈ 𝑠))
8		difeq1 4069	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑢 ∖ 𝑣) = (𝑥 ∖ 𝑣))
9	8	eleq1d 2819	. . . . . . . . . . . 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 4112	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑥 ∪ 𝑣) = (𝑥 ∪ 𝑦))
12	11	eleq1d 2819	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → ((𝑥 ∪ 𝑣) ∈ 𝑠 ↔ (𝑥 ∪ 𝑦) ∈ 𝑠))
13		difeq2 4070	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑥 ∖ 𝑣) = (𝑥 ∖ 𝑦))
14	13	eleq1d 2819	. . . . . . . . . . . 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 3216	. . . . . . . . . 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 3402	. . . . . . . 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 2760	. . . . . . 7 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 ((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠))}
20	19	inelros 34279	. . . . . 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 34278	. . . . . . . . 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 4763	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ⊆ 𝑆)
25		snex 5379	. . . . . . . 8 ⊢ {(𝑥 ∖ 𝑦)} ∈ V
26	25	elpw 4556	. . . . . . 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 8978	. . . . . . 7 ⊢ {(𝑥 ∖ 𝑦)} ∈ Fin
29	28	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ∈ Fin)
30		disjxsn 5090	. . . . . . 7 ⊢ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡
31	30	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡)
32		unisng 4879	. . . . . . . 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 2740	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∖ 𝑦) = ∪ {(𝑥 ∖ 𝑦)})
35		eleq1 2822	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → (𝑧 ∈ Fin ↔ {(𝑥 ∖ 𝑦)} ∈ Fin))
36		disjeq1 5070	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → (Disj 𝑡 ∈ 𝑧 𝑡 ↔ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡))
37		unieq 4872	. . . . . . . . 9 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → ∪ 𝑧 = ∪ {(𝑥 ∖ 𝑦)})
38	37	eqeq2d 2745	. . . . . . . 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 3574	. . . . . 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 3177	. . 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 34281	. 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 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  {crab 3397   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  𝒫 cpw 4552  {csn 4578  ∪ cuni 4861  Disj wdisj 5063  Fincfn 8881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-disj 5064  df-br 5097  df-opab 5159  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-om 7807  df-1o 8395  df-en 8882  df-fin 8885

This theorem is referenced by: (None)