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Theorem salunicl 41321
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 41319 . . . . . 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 224 . . . 4 (𝜑 → (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
76simp3d 1178 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))
8 breq1 4878 . . . . 5 (𝑦 = 𝑇 → (𝑦 ≼ ω ↔ 𝑇 ≼ ω))
9 unieq 4668 . . . . . 6 (𝑦 = 𝑇 𝑦 = 𝑇)
109eleq1d 2891 . . . . 5 (𝑦 = 𝑇 → ( 𝑦𝑆 𝑇𝑆))
118, 10imbi12d 336 . . . 4 (𝑦 = 𝑇 → ((𝑦 ≼ ω → 𝑦𝑆) ↔ (𝑇 ≼ ω → 𝑇𝑆)))
1211rspcva 3524 . . 3 ((𝑇 ∈ 𝒫 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)) → (𝑇 ≼ ω → 𝑇𝑆))
132, 7, 12syl2anc 579 . 2 (𝜑 → (𝑇 ≼ ω → 𝑇𝑆))
141, 13mpd 15 1 (𝜑 𝑇𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1111   = wceq 1656  wcel 2164  wral 3117  cdif 3795  c0 4146  𝒫 cpw 4380   cuni 4660   class class class wbr 4875  ωcom 7331  cdom 8226  SAlgcsalg 41313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  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-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-salg 41314
This theorem is referenced by:  saliuncl  41327  intsal  41333  smfpimbor1lem1  41793
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