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Theorem issal 42798
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 2904 . . 3 (𝑥 = 𝑆 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆))
2 id 22 . . . 4 (𝑥 = 𝑆𝑥 = 𝑆)
3 unieq 4830 . . . . . 6 (𝑥 = 𝑆 𝑥 = 𝑆)
43difeq1d 4082 . . . . 5 (𝑥 = 𝑆 → ( 𝑥𝑦) = ( 𝑆𝑦))
54, 2eleq12d 2910 . . . 4 (𝑥 = 𝑆 → (( 𝑥𝑦) ∈ 𝑥 ↔ ( 𝑆𝑦) ∈ 𝑆))
62, 5raleqbidv 3392 . . 3 (𝑥 = 𝑆 → (∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ↔ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆))
7 pweq 4536 . . . 4 (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
8 eleq2 2904 . . . . 5 (𝑥 = 𝑆 → ( 𝑦𝑥 𝑦𝑆))
98imbi2d 344 . . . 4 (𝑥 = 𝑆 → ((𝑦 ≼ ω → 𝑦𝑥) ↔ (𝑦 ≼ ω → 𝑦𝑆)))
107, 9raleqbidv 3392 . . 3 (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
111, 6, 103anbi123d 1433 . 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 42793 . 2 SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 ( 𝑥𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → 𝑦𝑥))}
1311, 12elab2g 3653 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 209  w3a 1084   = wceq 1538  wcel 2115  wral 3132  cdif 3915  c0 4274  𝒫 cpw 4520   cuni 4819   class class class wbr 5047  ωcom 7563  cdom 8490  SAlgcsalg 42792
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rab 3141  df-v 3481  df-dif 3921  df-in 3925  df-ss 3935  df-pw 4522  df-uni 4820  df-salg 42793
This theorem is referenced by:  pwsal  42799  salunicl  42800  saluncl  42801  prsal  42802  saldifcl  42803  0sal  42804  intsal  42812  issald  42815  caragensal  43006
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