Theorem saldifcld 43886

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 43860	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2106   ∖ cdif 3884  ∪ cuni 4839  SAlgcsalg 43849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-pw 4535  df-uni 4840  df-salg 43850

This theorem is referenced by:  subsalsal  43898  salpreimagelt  44244  salpreimalegt  44246  smfresal  44322