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Theorem issald 44727
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 4098 . . . 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 1121 . . 3 ((𝜑𝑦 ∈ 𝒫 𝑆) → (𝑦 ≼ ω → 𝑦𝑆))
109ralrimiva 3145 . 2 (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))
11 issald.s . . 3 (𝜑𝑆𝑉)
12 issal 44708 . . 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 1342 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3060  cdif 3925  c0 4302  𝒫 cpw 4580   cuni 4885   class class class wbr 5125  ωcom 7822  cdom 8903  SAlgcsalg 44702
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 2702
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 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3419  df-v 3461  df-dif 3931  df-in 3935  df-ss 3945  df-pw 4582  df-uni 4886  df-salg 44703
This theorem is referenced by:  salexct  44728  issalnnd  44739
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