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Theorem 0sald 46370
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 46340 . 2 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
31, 2syl 17 1 (𝜑 → ∅ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  c0 4332  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:  subsalsal  46379  smfpimltxr  46767  smfconst  46769  smfpimgtxr  46800  smfresal  46808
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