Theorem saldifcl 46275

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 4130	. . 3 ⊢ (𝑦 = 𝐸 → (∪ 𝑆 ∖ 𝑦) = (∪ 𝑆 ∖ 𝐸))
2	1	eleq1d 2824	. 2 ⊢ (𝑦 = 𝐸 → ((∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝐸) ∈ 𝑆))
3		issal 46270	. . . . 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 1142	. . 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 3623	1 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ∖ cdif 3960  ∅c0 4339  𝒫 cpw 4605  ∪ cuni 4912   class class class wbr 5148  ωcom 7887   ≼ cdom 8982  SAlgcsalg 46264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-ss 3980  df-pw 4607  df-uni 4913  df-salg 46265

This theorem is referenced by:  salincl  46280  saluni  46281  saliinclf  46282  saldifcl2  46284  intsal  46286  saldifcld  46303