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Theorem saluni 42953
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 4289 . 2 ( 𝑆 ∖ ∅) = 𝑆
2 0sal 42949 . . 3 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3 saldifcl 42948 . . 3 ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → ( 𝑆 ∖ ∅) ∈ 𝑆)
42, 3mpdan 686 . 2 (𝑆 ∈ SAlg → ( 𝑆 ∖ ∅) ∈ 𝑆)
51, 4eqeltrrid 2898 1 (𝑆 ∈ SAlg → 𝑆𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2112  cdif 3881  c0 4246   cuni 4803  SAlgcsalg 42937
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rab 3118  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247  df-pw 4502  df-uni 4804  df-salg 42938
This theorem is referenced by:  intsaluni  42956  unisalgen  42967  salgencntex  42970  salunid  42980
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