Theorem saluni 47474

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 4329	. 2 ⊢ (∪ 𝑆 ∖ ∅) = ∪ 𝑆
2		0sal 47469	. . 3 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3		saldifcl 47468	. . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → (∪ 𝑆 ∖ ∅) ∈ 𝑆)
4	2, 3	mpdan 700	. 2 ⊢ (𝑆 ∈ SAlg → (∪ 𝑆 ∖ ∅) ∈ 𝑆)
5	1, 4	eqeltrrid 2868	1 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ∖ cdif 4047  ∅c0 4280  ∪ cuni 4869  SAlgcsalg 47457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-nul 4281  df-pw 4561  df-uni 4870  df-salg 47458

This theorem is used by:  intsaluni  47478  unisalgen  47489  salgencntex  47492  salunid  47502