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Theorem issald 42815
 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 2833 . . . . 5 𝑆 = 𝑋
43difeq1i 4079 . . . 4 ( 𝑆𝑦) = (𝑋𝑦)
5 issald.d . . . 4 ((𝜑𝑦𝑆) → (𝑋𝑦) ∈ 𝑆)
64, 5eqeltrid 2920 . . 3 ((𝜑𝑦𝑆) → ( 𝑆𝑦) ∈ 𝑆)
76ralrimiva 3176 . 2 (𝜑 → ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆)
8 issald.u . . . 4 ((𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω) → 𝑦𝑆)
983expia 1118 . . 3 ((𝜑𝑦 ∈ 𝒫 𝑆) → (𝑦 ≼ ω → 𝑦𝑆))
109ralrimiva 3176 . 2 (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))
11 issald.s . . 3 (𝜑𝑆𝑉)
12 issal 42798 . . 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 1339 1 (𝜑𝑆 ∈ SAlg)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ 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:  salexct  42816  issalnnd  42827
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