Theorem rossros 31441

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 31430	. . . 4 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ⊆ 𝒫 𝑂)
3		elpwg 4544	. . . 4 ⊢ (𝑆 ∈ 𝑄 → (𝑆 ∈ 𝒫 𝒫 𝑂 ↔ 𝑆 ⊆ 𝒫 𝑂))
4	2, 3	mpbird 259	. . 3 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝒫 𝒫 𝑂)
5	1	0elros 31431	. . 3 ⊢ (𝑆 ∈ 𝑄 → ∅ ∈ 𝑆)
6		uneq1 4134	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑢 ∪ 𝑣) = (𝑥 ∪ 𝑣))
7	6	eleq1d 2899	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑥 → ((𝑢 ∪ 𝑣) ∈ 𝑠 ↔ (𝑥 ∪ 𝑣) ∈ 𝑠))
8		difeq1 4094	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑥 → (𝑢 ∖ 𝑣) = (𝑥 ∖ 𝑣))
9	8	eleq1d 2899	. . . . . . . . . . . 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 4135	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑥 ∪ 𝑣) = (𝑥 ∪ 𝑦))
12	11	eleq1d 2899	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → ((𝑥 ∪ 𝑣) ∈ 𝑠 ↔ (𝑥 ∪ 𝑦) ∈ 𝑠))
13		difeq2 4095	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑥 ∖ 𝑣) = (𝑥 ∖ 𝑦))
14	13	eleq1d 2899	. . . . . . . . . . . 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 3463	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 ((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠) ↔ ∀𝑥 ∈ 𝑠 ∀𝑦 ∈ 𝑠 ((𝑥 ∪ 𝑦) ∈ 𝑠 ∧ (𝑥 ∖ 𝑦) ∈ 𝑠))
17	16	anbi2i 624	. . . . . . . . 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 3475	. . . . . . . 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 2849	. . . . . . 7 ⊢ 𝑄 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 ((𝑢 ∪ 𝑣) ∈ 𝑠 ∧ (𝑢 ∖ 𝑣) ∈ 𝑠))}
20	19	inelros 31434	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∩ 𝑦) ∈ 𝑆)
21	20	3expb 1116	. . . . 5 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∩ 𝑦) ∈ 𝑆)
22	19	difelros 31433	. . . . . . . . 9 ⊢ ((𝑆 ∈ 𝑄 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∖ 𝑦) ∈ 𝑆)
23	22	3expb 1116	. . . . . . . 8 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∖ 𝑦) ∈ 𝑆)
24	23	snssd 4744	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ⊆ 𝑆)
25		snex 5334	. . . . . . . 8 ⊢ {(𝑥 ∖ 𝑦)} ∈ V
26	25	elpw 4545	. . . . . . 7 ⊢ ({(𝑥 ∖ 𝑦)} ∈ 𝒫 𝑆 ↔ {(𝑥 ∖ 𝑦)} ⊆ 𝑆)
27	24, 26	sylibr 236	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ∈ 𝒫 𝑆)
28		snfi 8596	. . . . . . 7 ⊢ {(𝑥 ∖ 𝑦)} ∈ Fin
29	28	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → {(𝑥 ∖ 𝑦)} ∈ Fin)
30		disjxsn 5061	. . . . . . 7 ⊢ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡
31	30	a1i 11	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡)
32		unisng 4859	. . . . . . . 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 2829	. . . . . 6 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 ∖ 𝑦) = ∪ {(𝑥 ∖ 𝑦)})
35		eleq1 2902	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → (𝑧 ∈ Fin ↔ {(𝑥 ∖ 𝑦)} ∈ Fin))
36		disjeq1 5040	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → (Disj 𝑡 ∈ 𝑧 𝑡 ↔ Disj 𝑡 ∈ {(𝑥 ∖ 𝑦)}𝑡))
37		unieq 4851	. . . . . . . . 9 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → ∪ 𝑧 = ∪ {(𝑥 ∖ 𝑦)})
38	37	eqeq2d 2834	. . . . . . . 8 ⊢ (𝑧 = {(𝑥 ∖ 𝑦)} → ((𝑥 ∖ 𝑦) = ∪ 𝑧 ↔ (𝑥 ∖ 𝑦) = ∪ {(𝑥 ∖ 𝑦)}))
39	35, 36, 38	3anbi123d 1432	. . . . . . 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 3625	. . . . . 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 1368	. . . . 5 ⊢ ((𝑆 ∈ 𝑄 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧))
42	21, 41	jca 514	. . . 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 1124	. 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 31436	. 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 236	1 ⊢ (𝑆 ∈ 𝑄 → 𝑆 ∈ 𝑁)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  {crab 3144   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  𝒫 cpw 4541  {csn 4569  ∪ cuni 4840  Disj wdisj 5033  Fincfn 8511

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-disj 5034  df-br 5069  df-opab 5131  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-om 7583  df-1o 8104  df-en 8512  df-fin 8515

This theorem is referenced by: (None)