Theorem saluni 44719

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 4352	. 2 ⊢ (∪ 𝑆 ∖ ∅) = ∪ 𝑆
2		0sal 44714	. . 3 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3		saldifcl 44713	. . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → (∪ 𝑆 ∖ ∅) ∈ 𝑆)
4	2, 3	mpdan 685	. 2 ⊢ (𝑆 ∈ SAlg → (∪ 𝑆 ∖ ∅) ∈ 𝑆)
5	1, 4	eqeltrrid 2837	1 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2106   ∖ cdif 3925  ∅c0 4302  ∪ cuni 4885  SAlgcsalg 44702

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3419  df-v 3461  df-dif 3931  df-in 3935  df-ss 3945  df-nul 4303  df-pw 4582  df-uni 4886  df-salg 44703

This theorem is referenced by:  intsaluni  44723  unisalgen  44734  salgencntex  44737  salunid  44747