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Theorem issal 41051
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 2839 . . 3 (𝑥 = 𝑆 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆))
2 id 22 . . . 4 (𝑥 = 𝑆𝑥 = 𝑆)
3 unieq 4582 . . . . . 6 (𝑥 = 𝑆 𝑥 = 𝑆)
43difeq1d 3878 . . . . 5 (𝑥 = 𝑆 → ( 𝑥𝑦) = ( 𝑆𝑦))
54, 2eleq12d 2844 . . . 4 (𝑥 = 𝑆 → (( 𝑥𝑦) ∈ 𝑥 ↔ ( 𝑆𝑦) ∈ 𝑆))
62, 5raleqbidv 3301 . . 3 (𝑥 = 𝑆 → (∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ↔ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆))
7 pweq 4300 . . . 4 (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
8 eleq2 2839 . . . . 5 (𝑥 = 𝑆 → ( 𝑦𝑥 𝑦𝑆))
98imbi2d 329 . . . 4 (𝑥 = 𝑆 → ((𝑦 ≼ ω → 𝑦𝑥) ↔ (𝑦 ≼ ω → 𝑦𝑆)))
107, 9raleqbidv 3301 . . 3 (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
111, 6, 103anbi123d 1547 . 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 41046 . 2 SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥))}
1311, 12elab2g 3504 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 196  w3a 1071   = wceq 1631  wcel 2145  wral 3061  cdif 3720  c0 4063  𝒫 cpw 4297   cuni 4574   class class class wbr 4786  ωcom 7212  cdom 8107  SAlgcsalg 41045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-in 3730  df-ss 3737  df-pw 4299  df-uni 4575  df-salg 41046
This theorem is referenced by:  pwsal  41052  salunicl  41053  saluncl  41054  prsal  41055  saldifcl  41056  0sal  41057  intsal  41065  issald  41068  caragensal  41259
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