Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  saluni Structured version   Visualization version   GIF version

Theorem saluni 41045
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 4091 . 2 ( 𝑆 ∖ ∅) = 𝑆
2 0sal 41041 . . 3 (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3 saldifcl 41040 . . 3 ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → ( 𝑆 ∖ ∅) ∈ 𝑆)
42, 3mpdan 705 . 2 (𝑆 ∈ SAlg → ( 𝑆 ∖ ∅) ∈ 𝑆)
51, 4syl5eqelr 2842 1 (𝑆 ∈ SAlg → 𝑆𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2137  cdif 3710  c0 4056   cuni 4586  SAlgcsalg 41029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-in 3720  df-ss 3727  df-nul 4057  df-pw 4302  df-uni 4587  df-salg 41030
This theorem is referenced by:  intsaluni  41048  unisalgen  41059  salgencntex  41062  salunid  41072
  Copyright terms: Public domain W3C validator