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Theorem saluni 39220
 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 3904 . . . 4 ( 𝑆 ∖ ∅) = 𝑆
21eqcomi 2619 . . 3 𝑆 = ( 𝑆 ∖ ∅)
32a1i 11 . 2 (𝑆 ∈ SAlg → 𝑆 = ( 𝑆 ∖ ∅))
4 id 22 . . 3 (𝑆 ∈ SAlg → 𝑆 ∈ SAlg)
5 0sal 39216 . . 3 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
6 saldifcl 39215 . . 3 ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → ( 𝑆 ∖ ∅) ∈ 𝑆)
74, 5, 6syl2anc 691 . 2 (𝑆 ∈ SAlg → ( 𝑆 ∖ ∅) ∈ 𝑆)
83, 7eqeltrd 2688 1 (𝑆 ∈ SAlg → 𝑆𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ∅c0 3874  ∪ cuni 4372  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-nul 3875  df-pw 4110  df-uni 4373  df-salg 39205 This theorem is referenced by:  intsaluni  39223  unisalgen  39234  salgencntex  39237  salunid  39247
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