Theorem inelsros 34509

Description: A semiring of sets is closed under union. (Contributed by Thierry Arnoux, 18-Jul-2020.)

Hypothesis

Ref	Expression
issros.1	⊢ 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
inelsros	⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)

Distinct variable groups:   𝑡,𝑠,𝑥,𝑦   𝑂,𝑠   𝑆,𝑠,𝑥,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧

Allowed substitution hints:   𝐴(𝑡,𝑠)   𝐵(𝑥,𝑡,𝑠)   𝑆(𝑡)   𝑁(𝑥,𝑦,𝑧,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑧,𝑡)

Proof of Theorem inelsros

Step	Hyp	Ref	Expression
1		simp2 1153	. . 3 ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆)
2		simp3 1154	. . 3 ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐵 ∈ 𝑆)
3		issros.1	. . . . . 6 ⊢ 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)))}
4	3	issros 34506	. . . . 5 ⊢ (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))))
5	4	simp3bi 1163	. . . 4 ⊢ (𝑆 ∈ 𝑁 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))
6	5	3ad2ant1 1149	. . 3 ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)))
7		ineq1 4174	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
8	7	eleq1d 2854	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∈ 𝑆 ↔ (𝐴 ∩ 𝑦) ∈ 𝑆))
9		difeq1 4082	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∖ 𝑦) = (𝐴 ∖ 𝑦))
10	9	eqeq1d 2771	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∖ 𝑦) = ∪ 𝑧 ↔ (𝐴 ∖ 𝑦) = ∪ 𝑧))
11	10	3anbi3d 1468	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧) ↔ (𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝑦) = ∪ 𝑧)))
12	11	rexbidv 3195	. . . . 5 ⊢ (𝑥 = 𝐴 → (∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧) ↔ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝑦) = ∪ 𝑧)))
13	8, 12	anbi12d 643	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧)) ↔ ((𝐴 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝑦) = ∪ 𝑧))))
14		ineq2 4175	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
15	14	eleq1d 2854	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∈ 𝑆 ↔ (𝐴 ∩ 𝐵) ∈ 𝑆))
16		difeq2 4083	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝐵))
17	16	eqeq1d 2771	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 ∖ 𝑦) = ∪ 𝑧 ↔ (𝐴 ∖ 𝐵) = ∪ 𝑧))
18	17	3anbi3d 1468	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝑦) = ∪ 𝑧) ↔ (𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧)))
19	18	rexbidv 3195	. . . . 5 ⊢ (𝑦 = 𝐵 → (∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝑦) = ∪ 𝑧) ↔ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧)))
20	15, 19	anbi12d 643	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝐴 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝑦) = ∪ 𝑧)) ↔ ((𝐴 ∩ 𝐵) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧))))
21	13, 20	rspc2va 3602	. . 3 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ((𝑥 ∩ 𝑦) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧))) → ((𝐴 ∩ 𝐵) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧)))
22	1, 2, 6, 21	syl21anc 850	. 2 ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴 ∩ 𝐵) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧)))
23	22	simpld 499	1 ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∩ 𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423   ∖ cdif 3910   ∩ cin 3912  ∅c0 4294  𝒫 cpw 4564  ∪ cuni 4873  Disj wdisj 5077  Fincfn 8939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-in 3920  df-ss 3930  df-pw 4566

This theorem is referenced by: (None)