Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  issal Structured version   Visualization version   GIF version

Theorem issal 41103
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 2833 . . 3 (𝑥 = 𝑆 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆))
2 id 22 . . . 4 (𝑥 = 𝑆𝑥 = 𝑆)
3 unieq 4602 . . . . . 6 (𝑥 = 𝑆 𝑥 = 𝑆)
43difeq1d 3889 . . . . 5 (𝑥 = 𝑆 → ( 𝑥𝑦) = ( 𝑆𝑦))
54, 2eleq12d 2838 . . . 4 (𝑥 = 𝑆 → (( 𝑥𝑦) ∈ 𝑥 ↔ ( 𝑆𝑦) ∈ 𝑆))
62, 5raleqbidv 3300 . . 3 (𝑥 = 𝑆 → (∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ↔ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆))
7 pweq 4318 . . . 4 (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
8 eleq2 2833 . . . . 5 (𝑥 = 𝑆 → ( 𝑦𝑥 𝑦𝑆))
98imbi2d 331 . . . 4 (𝑥 = 𝑆 → ((𝑦 ≼ ω → 𝑦𝑥) ↔ (𝑦 ≼ ω → 𝑦𝑆)))
107, 9raleqbidv 3300 . . 3 (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
111, 6, 103anbi123d 1560 . 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 41098 . 2 SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥))}
1311, 12elab2g 3508 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 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
  Copyright terms: Public domain W3C validator