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Theorem saluni 41039
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 4085 . 2 ( 𝑆 ∖ ∅) = 𝑆
2 0sal 41035 . . 3 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3 saldifcl 41034 . . 3 ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → ( 𝑆 ∖ ∅) ∈ 𝑆)
42, 3mpdan 705 . 2 (𝑆 ∈ SAlg → ( 𝑆 ∖ ∅) ∈ 𝑆)
51, 4syl5eqelr 2836 1 (𝑆 ∈ SAlg → 𝑆𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2131  cdif 3704  c0 4050   cuni 4580  SAlgcsalg 41023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-in 3714  df-ss 3721  df-nul 4051  df-pw 4296  df-uni 4581  df-salg 41024
This theorem is referenced by:  intsaluni  41042  unisalgen  41053  salgencntex  41056  salunid  41066
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