Theorem saldifcld 46329

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 46301	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 2109   ∖ cdif 3902  ∪ cuni 4861  SAlgcsalg 46290

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3397  df-v 3440  df-dif 3908  df-ss 3922  df-pw 4555  df-uni 4862  df-salg 46291

This theorem is referenced by:  subsalsal  46341  salpreimagelt  46689  salpreimalegt  46691  smfresal  46770