Theorem issal 42956

Description: Express the predicate "𝑆 is a sigma-algebra." (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
issal	⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))

Distinct variable group:   𝑦,𝑆

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem issal

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2w 2873	. . 3 ⊢ (𝑥 = 𝑧 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑧))
2		id 22	. . . 4 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
3		unieq 4811	. . . . . 6 ⊢ (𝑥 = 𝑧 → ∪ 𝑥 = ∪ 𝑧)
4	3	difeq1d 4049	. . . . 5 ⊢ (𝑥 = 𝑧 → (∪ 𝑥 ∖ 𝑦) = (∪ 𝑧 ∖ 𝑦))
5	4, 2	eleq12d 2884	. . . 4 ⊢ (𝑥 = 𝑧 → ((∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ (∪ 𝑧 ∖ 𝑦) ∈ 𝑧))
6	2, 5	raleqbidv 3354	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑧 (∪ 𝑧 ∖ 𝑦) ∈ 𝑧))
7		pweq 4513	. . . 4 ⊢ (𝑥 = 𝑧 → 𝒫 𝑥 = 𝒫 𝑧)
8		eleq2w 2873	. . . . 5 ⊢ (𝑥 = 𝑧 → (∪ 𝑦 ∈ 𝑥 ↔ ∪ 𝑦 ∈ 𝑧))
9	8	imbi2d 344	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑧)))
10	7, 9	raleqbidv 3354	. . 3 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑧(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑧)))
11	1, 6, 10	3anbi123d 1433	. 2 ⊢ (𝑥 = 𝑧 → ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥)) ↔ (∅ ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (∪ 𝑧 ∖ 𝑦) ∈ 𝑧 ∧ ∀𝑦 ∈ 𝒫 𝑧(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑧))))
12		eleq2 2878	. . 3 ⊢ (𝑧 = 𝑆 → (∅ ∈ 𝑧 ↔ ∅ ∈ 𝑆))
13		id 22	. . . 4 ⊢ (𝑧 = 𝑆 → 𝑧 = 𝑆)
14		unieq 4811	. . . . . 6 ⊢ (𝑧 = 𝑆 → ∪ 𝑧 = ∪ 𝑆)
15	14	difeq1d 4049	. . . . 5 ⊢ (𝑧 = 𝑆 → (∪ 𝑧 ∖ 𝑦) = (∪ 𝑆 ∖ 𝑦))
16	15, 13	eleq12d 2884	. . . 4 ⊢ (𝑧 = 𝑆 → ((∪ 𝑧 ∖ 𝑦) ∈ 𝑧 ↔ (∪ 𝑆 ∖ 𝑦) ∈ 𝑆))
17	13, 16	raleqbidv 3354	. . 3 ⊢ (𝑧 = 𝑆 → (∀𝑦 ∈ 𝑧 (∪ 𝑧 ∖ 𝑦) ∈ 𝑧 ↔ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆))
18		pweq 4513	. . . 4 ⊢ (𝑧 = 𝑆 → 𝒫 𝑧 = 𝒫 𝑆)
19		eleq2 2878	. . . . 5 ⊢ (𝑧 = 𝑆 → (∪ 𝑦 ∈ 𝑧 ↔ ∪ 𝑦 ∈ 𝑆))
20	19	imbi2d 344	. . . 4 ⊢ (𝑧 = 𝑆 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑧) ↔ (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
21	18, 20	raleqbidv 3354	. . 3 ⊢ (𝑧 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑧(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
22	12, 17, 21	3anbi123d 1433	. 2 ⊢ (𝑧 = 𝑆 → ((∅ ∈ 𝑧 ∧ ∀𝑦 ∈ 𝑧 (∪ 𝑧 ∖ 𝑦) ∈ 𝑧 ∧ ∀𝑦 ∈ 𝒫 𝑧(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑧)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
23		df-salg 42951	. 2 ⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))}
24	11, 22, 23	elab2gw 3613	1 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∖ cdif 3878  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4800   class class class wbr 5030  ωcom 7560   ≼ cdom 8490  SAlgcsalg 42950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-pw 4499  df-uni 4801  df-salg 42951

This theorem is referenced by:  pwsal  42957  salunicl  42958  saluncl  42959  prsal  42960  saldifcl  42961  0sal  42962  intsal  42970  issald  42973  caragensal  43164