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Theorem salunicl 41327
 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 41325 . . . . . 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 1180 . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))
8 breq1 4876 . . . . 5 (𝑦 = 𝑇 → (𝑦 ≼ ω ↔ 𝑇 ≼ ω))
9 unieq 4666 . . . . . 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 581 . 2 (𝜑 → (𝑇 ≼ ω → 𝑇𝑆))
141, 13mpd 15 1 (𝜑 𝑇𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3117   ∖ cdif 3795  ∅c0 4144  𝒫 cpw 4378  ∪ cuni 4658   class class class wbr 4873  ωcom 7326   ≼ cdom 8220  SAlgcsalg 41319 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  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 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-salg 41320 This theorem is referenced by:  saliuncl  41333  intsal  41339  smfpimbor1lem1  41799
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