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Theorem saluni 46340
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 4378 . 2 ( 𝑆 ∖ ∅) = 𝑆
2 0sal 46335 . . 3 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3 saldifcl 46334 . . 3 ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → ( 𝑆 ∖ ∅) ∈ 𝑆)
42, 3mpdan 687 . 2 (𝑆 ∈ SAlg → ( 𝑆 ∖ ∅) ∈ 𝑆)
51, 4eqeltrrid 2846 1 (𝑆 ∈ SAlg → 𝑆𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  cdif 3948  c0 4333   cuni 4907  SAlgcsalg 46323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334  df-pw 4602  df-uni 4908  df-salg 46324
This theorem is referenced by:  intsaluni  46344  unisalgen  46355  salgencntex  46358  salunid  46368
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