Theorem saluni 42603

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 → ∪ 𝑆 ∈ 𝑆)

Proof of Theorem saluni

Step	Hyp	Ref	Expression
1		dif0 4331	. 2 ⊢ (∪ 𝑆 ∖ ∅) = ∪ 𝑆
2		0sal 42599	. . 3 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3		saldifcl 42598	. . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → (∪ 𝑆 ∖ ∅) ∈ 𝑆)
4	2, 3	mpdan 685	. 2 ⊢ (𝑆 ∈ SAlg → (∪ 𝑆 ∖ ∅) ∈ 𝑆)
5	1, 4	eqeltrrid 2918	1 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2110   ∖ cdif 3932  ∅c0 4290  ∪ cuni 4831  SAlgcsalg 42587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-in 3942  df-ss 3951  df-nul 4291  df-pw 4540  df-uni 4832  df-salg 42588

This theorem is referenced by:  intsaluni  42606  unisalgen  42617  salgencntex  42620  salunid  42630