Theorem issal 42606

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 2903	. . 3 ⊢ (𝑥 = 𝑆 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆))
2		id 22	. . . 4 ⊢ (𝑥 = 𝑆 → 𝑥 = 𝑆)
3		unieq 4851	. . . . . 6 ⊢ (𝑥 = 𝑆 → ∪ 𝑥 = ∪ 𝑆)
4	3	difeq1d 4100	. . . . 5 ⊢ (𝑥 = 𝑆 → (∪ 𝑥 ∖ 𝑦) = (∪ 𝑆 ∖ 𝑦))
5	4, 2	eleq12d 2909	. . . 4 ⊢ (𝑥 = 𝑆 → ((∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ (∪ 𝑆 ∖ 𝑦) ∈ 𝑆))
6	2, 5	raleqbidv 3403	. . 3 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆))
7		pweq 4557	. . . 4 ⊢ (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
8		eleq2 2903	. . . . 5 ⊢ (𝑥 = 𝑆 → (∪ 𝑦 ∈ 𝑥 ↔ ∪ 𝑦 ∈ 𝑆))
9	8	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑆 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
10	7, 9	raleqbidv 3403	. . 3 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
11	1, 6, 10	3anbi123d 1432	. 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 42601	. 2 ⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))}
13	11, 12	elab2g 3670	1 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140   ∖ cdif 3935  ∅c0 4293  𝒫 cpw 4541  ∪ cuni 4840   class class class wbr 5068  ωcom 7582   ≼ cdom 8509  SAlgcsalg 42600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rab 3149  df-v 3498  df-dif 3941  df-in 3945  df-ss 3954  df-pw 4543  df-uni 4841  df-salg 42601

This theorem is referenced by:  pwsal  42607  salunicl  42608  saluncl  42609  prsal  42610  saldifcl  42611  0sal  42612  intsal  42620  issald  42623  caragensal  42814