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Theorem issald 41342
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 2834 . . . . 5 𝑆 = 𝑋
43difeq1i 3951 . . . 4 ( 𝑆𝑦) = (𝑋𝑦)
5 issald.d . . . 4 ((𝜑𝑦𝑆) → (𝑋𝑦) ∈ 𝑆)
64, 5syl5eqel 2910 . . 3 ((𝜑𝑦𝑆) → ( 𝑆𝑦) ∈ 𝑆)
76ralrimiva 3175 . 2 (𝜑 → ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆)
8 issald.u . . . 4 ((𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω) → 𝑦𝑆)
983expia 1156 . . 3 ((𝜑𝑦 ∈ 𝒫 𝑆) → (𝑦 ≼ ω → 𝑦𝑆))
109ralrimiva 3175 . 2 (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))
11 issald.s . . 3 (𝜑𝑆𝑉)
12 issal 41325 . . 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 1448 1 (𝜑𝑆 ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113   = wceq 1658  wcel 2166  wral 3117  cdif 3795  c0 4144  𝒫 cpw 4378   cuni 4658   class class class wbr 4873  ωcom 7326  cdom 8220  SAlgcsalg 41319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-in 3805  df-ss 3812  df-pw 4380  df-uni 4659  df-salg 41320
This theorem is referenced by:  salexct  41343  issalnnd  41354
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