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Theorem 0sal 43581
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 43575 . . 3 (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆))))
21ibi 270 . 2 (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦𝑆 ( 𝑆𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → 𝑦𝑆)))
32simp1d 1144 1 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089  wcel 2112  wral 3064  cdif 3880  c0 4253  𝒫 cpw 4529   cuni 4835   class class class wbr 5069  ωcom 7665  cdom 8647  SAlgcsalg 43569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rab 3073  df-v 3425  df-dif 3886  df-in 3890  df-ss 3900  df-pw 4531  df-uni 4836  df-salg 43570
This theorem is referenced by:  saluni  43585  intsal  43589  0sald  43609  ismeannd  43725
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