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Theorem salunicl 44710
Description: SAlg sigma-algebra is closed under countable union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
salunicl.s (𝜑𝑆 ∈ SAlg)
salunicl.t (𝜑𝑇 ∈ 𝒫 𝑆)
salunicl.tct (𝜑𝑇 ≼ ω)
Assertion
Ref Expression
salunicl (𝜑 𝑇𝑆)

Proof of Theorem salunicl
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 salunicl.tct . 2 (𝜑𝑇 ≼ ω)
2 breq1 5128 . . . 4 (𝑦 = 𝑇 → (𝑦 ≼ ω ↔ 𝑇 ≼ ω))
3 unieq 4896 . . . . 5 (𝑦 = 𝑇 𝑦 = 𝑇)
43eleq1d 2817 . . . 4 (𝑦 = 𝑇 → ( 𝑦𝑆 𝑇𝑆))
52, 4imbi12d 344 . . 3 (𝑦 = 𝑇 → ((𝑦 ≼ ω → 𝑦𝑆) ↔ (𝑇 ≼ ω → 𝑇𝑆)))
6 salunicl.s . . . . 5 (𝜑𝑆 ∈ SAlg)
7 issal 44708 . . . . . 6 (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
86, 7syl 17 . . . . 5 (𝜑 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
96, 8mpbid 231 . . . 4 (𝜑 → (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
109simp3d 1144 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))
11 salunicl.t . . 3 (𝜑𝑇 ∈ 𝒫 𝑆)
125, 10, 11rspcdva 3596 . 2 (𝜑 → (𝑇 ≼ ω → 𝑇𝑆))
131, 12mpd 15 1 (𝜑 𝑇𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  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-or 846  df-3an 1089  df-tru 1544  df-fal 1554  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-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-salg 44703
This theorem is referenced by:  saliunclf  44716  intsal  44724  smfpimbor1lem1  45192
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