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Theorem issal 44456
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 2826 . . 3 (𝑥 = 𝑆 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆))
2 id 22 . . . 4 (𝑥 = 𝑆𝑥 = 𝑆)
3 unieq 4874 . . . . . 6 (𝑥 = 𝑆 𝑥 = 𝑆)
43difeq1d 4079 . . . . 5 (𝑥 = 𝑆 → ( 𝑥𝑦) = ( 𝑆𝑦))
54, 2eleq12d 2832 . . . 4 (𝑥 = 𝑆 → (( 𝑥𝑦) ∈ 𝑥 ↔ ( 𝑆𝑦) ∈ 𝑆))
62, 5raleqbidv 3317 . . 3 (𝑥 = 𝑆 → (∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ↔ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆))
7 pweq 4572 . . . 4 (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
8 eleq2 2826 . . . . 5 (𝑥 = 𝑆 → ( 𝑦𝑥 𝑦𝑆))
98imbi2d 340 . . . 4 (𝑥 = 𝑆 → ((𝑦 ≼ ω → 𝑦𝑥) ↔ (𝑦 ≼ ω → 𝑦𝑆)))
107, 9raleqbidv 3317 . . 3 (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
111, 6, 103anbi123d 1436 . 2 (𝑥 = 𝑆 → ((∅ ∈ 𝑥 ∧ ∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
12 df-salg 44451 . 2 SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥))}
1311, 12elab2g 3630 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 205  w3a 1087   = wceq 1541  wcel 2106  wral 3062  cdif 3905  c0 4280  𝒫 cpw 4558   cuni 4863   class class class wbr 5103  ωcom 7794  cdom 8839  SAlgcsalg 44450
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rab 3406  df-v 3445  df-dif 3911  df-in 3915  df-ss 3925  df-pw 4560  df-uni 4864  df-salg 44451
This theorem is referenced by:  pwsal  44457  salunicl  44458  saluncl  44459  prsal  44460  saldifcl  44461  0sal  44462  intsal  44472  issald  44475  caragensal  44667
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