Theorem saldifcl 46776

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 4054	. . 3 ⊢ (𝑦 = 𝐸 → (∪ 𝑆 ∖ 𝑦) = (∪ 𝑆 ∖ 𝐸))
2	1	eleq1d 2826	. 2 ⊢ (𝑦 = 𝐸 → ((∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝐸) ∈ 𝑆))
3		issal 46771	. . . . 5 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
4	3	ibi 269	. . . 4 ⊢ (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
5	4	simp2d 1150	. . 3 ⊢ (𝑆 ∈ SAlg → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
6	5	adantr 482	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
7		simpr 486	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → 𝐸 ∈ 𝑆)
8	2, 6, 7	rspcdva 3563	1 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055   ∖ cdif 3882  ∅c0 4264  𝒫 cpw 4532  ∪ cuni 4841   class class class wbr 5075  ωcom 7810   ≼ cdom 8885  SAlgcsalg 46765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-3an 1095  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rab 3394  df-v 3435  df-dif 3888  df-ss 3902  df-pw 4534  df-uni 4842  df-salg 46766

This theorem is referenced by:  salincl  46781  saluni  46782  saliinclf  46783  saldifcl2  46785  intsal  46787  saldifcld  46804