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Theorem saluni 41282
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 4152 . 2 ( 𝑆 ∖ ∅) = 𝑆
2 0sal 41278 . . 3 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3 saldifcl 41277 . . 3 ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → ( 𝑆 ∖ ∅) ∈ 𝑆)
42, 3mpdan 679 . 2 (𝑆 ∈ SAlg → ( 𝑆 ∖ ∅) ∈ 𝑆)
51, 4syl5eqelr 2884 1 (𝑆 ∈ SAlg → 𝑆𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  cdif 3767  c0 4116   cuni 4629  SAlgcsalg 41266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-in 3777  df-ss 3784  df-nul 4117  df-pw 4352  df-uni 4630  df-salg 41267
This theorem is referenced by:  intsaluni  41285  unisalgen  41296  salgencntex  41299  salunid  41309
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