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Theorem issald 45348
Description: Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
issald.s (𝜑𝑆𝑉)
issald.z (𝜑 → ∅ ∈ 𝑆)
issald.x 𝑋 = 𝑆
issald.d ((𝜑𝑦𝑆) → (𝑋𝑦) ∈ 𝑆)
issald.u ((𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω) → 𝑦𝑆)
Assertion
Ref Expression
issald (𝜑𝑆 ∈ SAlg)
Distinct variable groups:   𝑦,𝑆   𝜑,𝑦
Allowed substitution hints:   𝑉(𝑦)   𝑋(𝑦)

Proof of Theorem issald
StepHypRef Expression
1 issald.z . 2 (𝜑 → ∅ ∈ 𝑆)
2 issald.x . . . . . 6 𝑋 = 𝑆
32eqcomi 2740 . . . . 5 𝑆 = 𝑋
43difeq1i 4118 . . . 4 ( 𝑆𝑦) = (𝑋𝑦)
5 issald.d . . . 4 ((𝜑𝑦𝑆) → (𝑋𝑦) ∈ 𝑆)
64, 5eqeltrid 2836 . . 3 ((𝜑𝑦𝑆) → ( 𝑆𝑦) ∈ 𝑆)
76ralrimiva 3145 . 2 (𝜑 → ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆)
8 issald.u . . . 4 ((𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω) → 𝑦𝑆)
983expia 1120 . . 3 ((𝜑𝑦 ∈ 𝒫 𝑆) → (𝑦 ≼ ω → 𝑦𝑆))
109ralrimiva 3145 . 2 (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))
11 issald.s . . 3 (𝜑𝑆𝑉)
12 issal 45329 . . 3 (𝑆𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
1311, 12syl 17 . 2 (𝜑 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
141, 7, 10, 13mpbir3and 1341 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  cdif 3945  c0 4322  𝒫 cpw 4602   cuni 4908   class class class wbr 5148  ωcom 7859  cdom 8941  SAlgcsalg 45323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3432  df-v 3475  df-dif 3951  df-in 3955  df-ss 3965  df-pw 4604  df-uni 4909  df-salg 45324
This theorem is referenced by:  salexct  45349  issalnnd  45360
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