Theorem saldifcld 47496

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 47468	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ∖ cdif 4047  ∪ cuni 4869  SAlgcsalg 47457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-pw 4561  df-uni 4870  df-salg 47458

This theorem is used by:  subsalsal  47508  salpreimagelt  47856  salpreimalegt  47858  smfresal  47937