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Theorem 0sal 46276
Description: The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
0sal (𝑆 ∈ SAlg → ∅ ∈ 𝑆)

Proof of Theorem 0sal
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 issal 46270 . . 3 (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
21ibi 267 . 2 (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
32simp1d 1141 1 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2106  wral 3059  cdif 3960  c0 4339  𝒫 cpw 4605   cuni 4912   class class class wbr 5148  ωcom 7887  cdom 8982  SAlgcsalg 46264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-ss 3980  df-pw 4607  df-uni 4913  df-salg 46265
This theorem is referenced by:  saluni  46281  intsal  46286  0sald  46306  ismeannd  46423
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