Theorem saldifcld 39241

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 39215	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)
4	1, 2, 3	syl2anc 691	1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Syntax hints:   → wi 4   ∈ wcel 1977   ∖ cdif 3537  ∪ cuni 4372  SAlgcsalg 39204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-salg 39205

This theorem is referenced by:  subsalsal  39253  salpreimagelt  39595  salpreimalegt  39597  smfresal  39673