Theorem saldifcl 47134

Description: The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
saldifcl	⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Proof of Theorem saldifcl

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		difeq2 4071	. . 3 ⊢ (𝑦 = 𝐸 → (∪ 𝑆 ∖ 𝑦) = (∪ 𝑆 ∖ 𝐸))
2	1	eleq1d 2847	. 2 ⊢ (𝑦 = 𝐸 → ((∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝐸) ∈ 𝑆))
3		issal 47129	. . . . 5 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
4	3	ibi 270	. . . 4 ⊢ (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
5	4	simp2d 1161	. . 3 ⊢ (𝑆 ∈ SAlg → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
6	5	adantr 486	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
7		simpr 490	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → 𝐸 ∈ 𝑆)
8	2, 6, 7	rspcdva 3580	1 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ∖ cdif 3899  ∅c0 4282  𝒫 cpw 4560  ∪ cuni 4870   class class class wbr 5107  ωcom 7865   ≼ cdom 8953  SAlgcsalg 47123

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-ss 3919  df-pw 4562  df-uni 4871  df-salg 47124

This theorem is used by:  salincl  47139  saluni  47140  saliinclf  47141  saldifcl2  47143  intsal  47145  saldifcld  47162