Theorem issal 41103

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 2833	. . 3 ⊢ (𝑥 = 𝑆 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆))
2		id 22	. . . 4 ⊢ (𝑥 = 𝑆 → 𝑥 = 𝑆)
3		unieq 4602	. . . . . 6 ⊢ (𝑥 = 𝑆 → ∪ 𝑥 = ∪ 𝑆)
4	3	difeq1d 3889	. . . . 5 ⊢ (𝑥 = 𝑆 → (∪ 𝑥 ∖ 𝑦) = (∪ 𝑆 ∖ 𝑦))
5	4, 2	eleq12d 2838	. . . 4 ⊢ (𝑥 = 𝑆 → ((∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ (∪ 𝑆 ∖ 𝑦) ∈ 𝑆))
6	2, 5	raleqbidv 3300	. . 3 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆))
7		pweq 4318	. . . 4 ⊢ (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
8		eleq2 2833	. . . . 5 ⊢ (𝑥 = 𝑆 → (∪ 𝑦 ∈ 𝑥 ↔ ∪ 𝑦 ∈ 𝑆))
9	8	imbi2d 331	. . . 4 ⊢ (𝑥 = 𝑆 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
10	7, 9	raleqbidv 3300	. . 3 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
11	1, 6, 10	3anbi123d 1560	. 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 41098	. 2 ⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))}
13	11, 12	elab2g 3508	1 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 197   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155  ∀wral 3055   ∖ cdif 3729  ∅c0 4079  𝒫 cpw 4315  ∪ cuni 4594   class class class wbr 4809  ωcom 7263   ≼ cdom 8158  SAlgcsalg 41097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-in 3739  df-ss 3746  df-pw 4317  df-uni 4595  df-salg 41098

This theorem is referenced by:  pwsal  41104  salunicl  41105  saluncl  41106  prsal  41107  saldifcl  41108  0sal  41109  intsal  41117  issald  41120  caragensal  41311