Theorem saldifcl 46320

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 2813	. 2 ⊢ (𝑦 = 𝐸 → ((∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝐸) ∈ 𝑆))
3		issal 46315	. . . . 5 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
4	3	ibi 267	. . . 4 ⊢ (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
5	4	simp2d 1143	. . 3 ⊢ (𝑆 ∈ SAlg → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
6	5	adantr 480	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
7		simpr 484	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → 𝐸 ∈ 𝑆)
8	2, 6, 7	rspcdva 3578	1 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∖ cdif 3900  ∅c0 4284  𝒫 cpw 4551  ∪ cuni 4858   class class class wbr 5092  ωcom 7799   ≼ cdom 8870  SAlgcsalg 46309

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 3395  df-v 3438  df-dif 3906  df-ss 3920  df-pw 4553  df-uni 4859  df-salg 46310

This theorem is referenced by:  salincl  46325  saluni  46326  saliinclf  46327  saldifcl2  46329  intsal  46331  saldifcld  46348