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Theorem issal 39210
 Description: Express the predicate "𝑆 is a sigma-algebra." (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
issal (𝑆𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
Distinct variable group:   𝑦,𝑆
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem issal
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq2 2677 . . 3 (𝑥 = 𝑆 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆))
2 raleq 3115 . . . 4 (𝑥 = 𝑆 → (∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ↔ ∀𝑦𝑆 ( 𝑥𝑦) ∈ 𝑥))
3 unieq 4380 . . . . . . 7 (𝑥 = 𝑆 𝑥 = 𝑆)
43difeq1d 3689 . . . . . 6 (𝑥 = 𝑆 → ( 𝑥𝑦) = ( 𝑆𝑦))
5 id 22 . . . . . 6 (𝑥 = 𝑆𝑥 = 𝑆)
64, 5eleq12d 2682 . . . . 5 (𝑥 = 𝑆 → (( 𝑥𝑦) ∈ 𝑥 ↔ ( 𝑆𝑦) ∈ 𝑆))
76ralbidv 2969 . . . 4 (𝑥 = 𝑆 → (∀𝑦𝑆 ( 𝑥𝑦) ∈ 𝑥 ↔ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆))
82, 7bitrd 267 . . 3 (𝑥 = 𝑆 → (∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ↔ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆))
9 pweq 4111 . . . . 5 (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
109raleqdv 3121 . . . 4 (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑥)))
11 eleq2 2677 . . . . . 6 (𝑥 = 𝑆 → ( 𝑦𝑥 𝑦𝑆))
1211imbi2d 329 . . . . 5 (𝑥 = 𝑆 → ((𝑦 ≼ ω → 𝑦𝑥) ↔ (𝑦 ≼ ω → 𝑦𝑆)))
1312ralbidv 2969 . . . 4 (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
1410, 13bitrd 267 . . 3 (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
151, 8, 143anbi123d 1391 . 2 (𝑥 = 𝑆 → ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
16 df-salg 39205 . 2 SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥))}
1715, 16elab2g 3322 1 (𝑆𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  ωcom 6957   ≼ cdom 7839  SAlgcsalg 39204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-salg 39205 This theorem is referenced by:  pwsal  39211  salunicl  39212  saluncl  39213  prsal  39214  saldifcl  39215  0sal  39216  intsal  39224  issald  39227  caragensal  39415
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