Theorem saldifcld 43776

Description: The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
saldifcld.1	⊢ (𝜑 → 𝑆 ∈ SAlg)
saldifcld.2	⊢ (𝜑 → 𝐸 ∈ 𝑆)

Assertion

Ref	Expression
saldifcld	⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Proof of Theorem saldifcld

Step	Hyp	Ref	Expression
1		saldifcld.1	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
2		saldifcld.2	. 2 ⊢ (𝜑 → 𝐸 ∈ 𝑆)
3		saldifcl 43750	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2108   ∖ cdif 3880  ∪ cuni 4836  SAlgcsalg 43739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-pw 4532  df-uni 4837  df-salg 43740

This theorem is referenced by:  subsalsal  43788  salpreimagelt  44132  salpreimalegt  44134  smfresal  44209