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Theorem 0sald 39244
 Description: The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
0sald.1 (𝜑𝑆 ∈ SAlg)
Assertion
Ref Expression
0sald (𝜑 → ∅ ∈ 𝑆)

Proof of Theorem 0sald
StepHypRef Expression
1 0sald.1 . 2 (𝜑𝑆 ∈ SAlg)
2 0sal 39216 . 2 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
31, 2syl 17 1 (𝜑 → ∅ ∈ 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ∅c0 3874  SAlgcsalg 39204 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-salg 39205 This theorem is referenced by:  subsalsal  39253  smfpimltxr  39634  smfconst  39636  smfpimgtxr  39666  smfresal  39673
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