Theorem diffiunisros 34794

Description: In semiring of sets, complements can be written as finite disjoint unions. (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
diffiunisros	⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧))

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 diffiunisros

Step	Hyp	Ref	Expression
1		simp2 1155	. . 3 ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆)
2		simp3 1156	. . 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 34790	. . . . 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 1165	. . . 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 1151	. . 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 4159	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
8	7	eleq1d 2846	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∈ 𝑆 ↔ (𝐴 ∩ 𝑦) ∈ 𝑆))
9		difeq1 4067	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∖ 𝑦) = (𝐴 ∖ 𝑦))
10	9	eqeq1d 2763	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∖ 𝑦) = ∪ 𝑧 ↔ (𝐴 ∖ 𝑦) = ∪ 𝑧))
11	10	3anbi3d 1470	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧) ↔ (𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝑦) = ∪ 𝑧)))
12	11	rexbidv 3187	. . . . 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 644	. . . 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 4160	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
15	14	eleq1d 2846	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∈ 𝑆 ↔ (𝐴 ∩ 𝐵) ∈ 𝑆))
16		difeq2 4068	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝐵))
17	16	eqeq1d 2763	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 ∖ 𝑦) = ∪ 𝑧 ↔ (𝐴 ∖ 𝐵) = ∪ 𝑧))
18	17	3anbi3d 1470	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝑦) = ∪ 𝑧) ↔ (𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧)))
19	18	rexbidv 3187	. . . . 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 644	. . . 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 3588	. . 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 851	. 2 ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴 ∩ 𝐵) ∈ 𝑆 ∧ ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧)))
23	22	simprd 501	1 ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {crab 3413   ∖ cdif 3896   ∩ cin 3898  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867  Disj wdisj 5070  Fincfn 8957

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-in 3906  df-ss 3916  df-pw 4559

This theorem is used by: (None)