Theorem issal 40534

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 variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2690	. . 3 ⊢ (𝑥 = 𝑆 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆))
2		id 22	. . . 4 ⊢ (𝑥 = 𝑆 → 𝑥 = 𝑆)
3		unieq 4444	. . . . . 6 ⊢ (𝑥 = 𝑆 → ∪ 𝑥 = ∪ 𝑆)
4	3	difeq1d 3727	. . . . 5 ⊢ (𝑥 = 𝑆 → (∪ 𝑥 ∖ 𝑦) = (∪ 𝑆 ∖ 𝑦))
5	4, 2	eleq12d 2695	. . . 4 ⊢ (𝑥 = 𝑆 → ((∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ (∪ 𝑆 ∖ 𝑦) ∈ 𝑆))
6	2, 5	raleqbidv 3152	. . 3 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆))
7		pweq 4161	. . . 4 ⊢ (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
8		eleq2 2690	. . . . 5 ⊢ (𝑥 = 𝑆 → (∪ 𝑦 ∈ 𝑥 ↔ ∪ 𝑦 ∈ 𝑆))
9	8	imbi2d 330	. . . 4 ⊢ (𝑥 = 𝑆 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
10	7, 9	raleqbidv 3152	. . 3 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
11	1, 6, 10	3anbi123d 1399	. 2 ⊢ (𝑥 = 𝑆 → ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
12		df-salg 40529	. 2 ⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))}
13	11, 12	elab2g 3353	1 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∖ cdif 3571  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436   class class class wbr 4653  ωcom 7065   ≼ cdom 7953  SAlgcsalg 40528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-salg 40529

This theorem is referenced by:  pwsal  40535  salunicl  40536  saluncl  40537  prsal  40538  saldifcl  40539  0sal  40540  intsal  40548  issald  40551  caragensal  40739