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Theorem saluni 44719
Description: A set is an element of any sigma-algebra on it. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Assertion
Ref Expression
saluni (𝑆 ∈ SAlg → 𝑆𝑆)

Proof of Theorem saluni
StepHypRef Expression
1 dif0 4352 . 2 ( 𝑆 ∖ ∅) = 𝑆
2 0sal 44714 . . 3 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3 saldifcl 44713 . . 3 ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → ( 𝑆 ∖ ∅) ∈ 𝑆)
42, 3mpdan 685 . 2 (𝑆 ∈ SAlg → ( 𝑆 ∖ ∅) ∈ 𝑆)
51, 4eqeltrrid 2837 1 (𝑆 ∈ SAlg → 𝑆𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cdif 3925  c0 4302   cuni 4885  SAlgcsalg 44702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3419  df-v 3461  df-dif 3931  df-in 3935  df-ss 3945  df-nul 4303  df-pw 4582  df-uni 4886  df-salg 44703
This theorem is referenced by:  intsaluni  44723  unisalgen  44734  salgencntex  44737  salunid  44747
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