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Theorem 0sal 46340
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 46334 . . 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 1142 1 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2107  wral 3060  cdif 3947  c0 4332  𝒫 cpw 4599   cuni 4906   class class class wbr 5142  ωcom 7888  cdom 8984  SAlgcsalg 46328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-ss 3967  df-pw 4601  df-uni 4907  df-salg 46329
This theorem is referenced by:  saluni  46345  intsal  46350  0sald  46370  ismeannd  46487
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