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Theorem salunicl 39869
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 salunicl.t . . 3 (𝜑𝑇 ∈ 𝒫 𝑆)
3 salunicl.s . . . . 5 (𝜑𝑆 ∈ SAlg)
4 issal 39867 . . . . . 6 (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
53, 4syl 17 . . . . 5 (𝜑 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
63, 5mpbid 222 . . . 4 (𝜑 → (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
76simp3d 1073 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))
8 breq1 4621 . . . . 5 (𝑦 = 𝑇 → (𝑦 ≼ ω ↔ 𝑇 ≼ ω))
9 unieq 4415 . . . . . 6 (𝑦 = 𝑇 𝑦 = 𝑇)
109eleq1d 2683 . . . . 5 (𝑦 = 𝑇 → ( 𝑦𝑆 𝑇𝑆))
118, 10imbi12d 334 . . . 4 (𝑦 = 𝑇 → ((𝑦 ≼ ω → 𝑦𝑆) ↔ (𝑇 ≼ ω → 𝑇𝑆)))
1211rspcva 3296 . . 3 ((𝑇 ∈ 𝒫 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)) → (𝑇 ≼ ω → 𝑇𝑆))
132, 7, 12syl2anc 692 . 2 (𝜑 → (𝑇 ≼ ω → 𝑇𝑆))
141, 13mpd 15 1 (𝜑 𝑇𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  wcel 1987  wral 2907  cdif 3556  c0 3896  𝒫 cpw 4135   cuni 4407   class class class wbr 4618  ωcom 7019  cdom 7905  SAlgcsalg 39861
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-salg 39862
This theorem is referenced by:  saliuncl  39875  intsal  39881  smfpimbor1lem1  40338
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