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Theorem 0sald 46306
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 46276 . 2 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
31, 2syl 17 1 (𝜑 → ∅ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  c0 4339  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:  subsalsal  46315  smfpimltxr  46703  smfconst  46705  smfpimgtxr  46736  smfresal  46744
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