Theorem diffiunisros 34370

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 1143	. . 3 ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝐴 ∈ 𝑆)
2		simp3 1144	. . 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 34366	. . . . 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 1153	. . . 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 1139	. . 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 4149	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∩ 𝑦) = (𝐴 ∩ 𝑦))
8	7	eleq1d 2825	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ∩ 𝑦) ∈ 𝑆 ↔ (𝐴 ∩ 𝑦) ∈ 𝑆))
9		difeq1 4057	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∖ 𝑦) = (𝐴 ∖ 𝑦))
10	9	eqeq1d 2742	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∖ 𝑦) = ∪ 𝑧 ↔ (𝐴 ∖ 𝑦) = ∪ 𝑧))
11	10	3anbi3d 1450	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝑥 ∖ 𝑦) = ∪ 𝑧) ↔ (𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝑦) = ∪ 𝑧)))
12	11	rexbidv 3164	. . . . 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 638	. . . 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 4150	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ∩ 𝑦) = (𝐴 ∩ 𝐵))
15	14	eleq1d 2825	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 ∩ 𝑦) ∈ 𝑆 ↔ (𝐴 ∩ 𝐵) ∈ 𝑆))
16		difeq2 4058	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 ∖ 𝑦) = (𝐴 ∖ 𝐵))
17	16	eqeq1d 2742	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 ∖ 𝑦) = ∪ 𝑧 ↔ (𝐴 ∖ 𝐵) = ∪ 𝑧))
18	17	3anbi3d 1450	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝑦) = ∪ 𝑧) ↔ (𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧)))
19	18	rexbidv 3164	. . . . 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 638	. . . 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 3579	. . 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 843	. 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 496	1 ⊢ ((𝑆 ∈ 𝑁 ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∃𝑧 ∈ 𝒫 𝑆(𝑧 ∈ Fin ∧ Disj 𝑡 ∈ 𝑧 𝑡 ∧ (𝐴 ∖ 𝐵) = ∪ 𝑧))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  {crab 3392   ∖ cdif 3887   ∩ cin 3889  ∅c0 4268  𝒫 cpw 4536  ∪ cuni 4845  Disj wdisj 5046  Fincfn 8890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-in 3897  df-ss 3907  df-pw 4538

This theorem is referenced by: (None)