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Theorem salunicl 44528
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 5108 . . . 4 (𝑦 = 𝑇 → (𝑦 ≼ ω ↔ 𝑇 ≼ ω))
3 unieq 4876 . . . . 5 (𝑦 = 𝑇 𝑦 = 𝑇)
43eleq1d 2822 . . . 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 44526 . . . . . 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 3582 . 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 3064  cdif 3907  c0 4282  𝒫 cpw 4560   cuni 4865   class class class wbr 5105  ωcom 7801  cdom 8880  SAlgcsalg 44520
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 2707
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 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-salg 44521
This theorem is referenced by:  saliunclf  44534  intsal  44542  smfpimbor1lem1  45010
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