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Theorem issald 40314
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 2629 . . . . 5 𝑆 = 𝑋
43difeq1i 3716 . . . 4 ( 𝑆𝑦) = (𝑋𝑦)
5 issald.d . . . 4 ((𝜑𝑦𝑆) → (𝑋𝑦) ∈ 𝑆)
64, 5syl5eqel 2703 . . 3 ((𝜑𝑦𝑆) → ( 𝑆𝑦) ∈ 𝑆)
76ralrimiva 2963 . 2 (𝜑 → ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆)
8 issald.u . . . 4 ((𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω) → 𝑦𝑆)
983expia 1265 . . 3 ((𝜑𝑦 ∈ 𝒫 𝑆) → (𝑦 ≼ ω → 𝑦𝑆))
109ralrimiva 2963 . 2 (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))
11 issald.s . . 3 (𝜑𝑆𝑉)
12 issal 40297 . . 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 1243 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  cdif 3564  c0 3907  𝒫 cpw 4149   cuni 4427   class class class wbr 4644  ωcom 7050  cdom 7938  SAlgcsalg 40291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-in 3574  df-ss 3581  df-pw 4151  df-uni 4428  df-salg 40292
This theorem is referenced by:  salexct  40315  issalnnd  40326
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