Theorem saluni 47156

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 4327	. 2 ⊢ (∪ 𝑆 ∖ ∅) = ∪ 𝑆
2		0sal 47151	. . 3 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆)
3		saldifcl 47150	. . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → (∪ 𝑆 ∖ ∅) ∈ 𝑆)
4	2, 3	mpdan 700	. 2 ⊢ (𝑆 ∈ SAlg → (∪ 𝑆 ∖ ∅) ∈ 𝑆)
5	1, 4	eqeltrrid 2865	1 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ∖ cdif 3896  ∅c0 4279  ∪ cuni 4867  SAlgcsalg 47139

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-ss 3916  df-nul 4280  df-pw 4559  df-uni 4868  df-salg 47140

This theorem is used by:  intsaluni  47160  unisalgen  47171  salgencntex  47174  salunid  47184