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Theorem 0sald 44681
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 44651 . 2 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
31, 2syl 17 1 (𝜑 → ∅ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  c0 4286  SAlgcsalg 44639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-in 3921  df-ss 3931  df-pw 4566  df-uni 4870  df-salg 44640
This theorem is referenced by:  subsalsal  44690  smfpimltxr  45078  smfconst  45080  smfpimgtxr  45111  smfresal  45119
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