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 → ∪ 𝑆 ∈ 𝑆)

Proof of Theorem saluni

Step	Hyp	Ref	Expression
1		dif0 4378	. 2 ⊢ (∪ 𝑆 ∖ ∅) = ∪ 𝑆
2		0sal 46335	. . 3 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3		saldifcl 46334	. . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → (∪ 𝑆 ∖ ∅) ∈ 𝑆)
4	2, 3	mpdan 687	. 2 ⊢ (𝑆 ∈ SAlg → (∪ 𝑆 ∖ ∅) ∈ 𝑆)
5	1, 4	eqeltrrid 2846	1 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆)

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