Theorem saldifcl 44713

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 4096	. . 3 ⊢ (𝑦 = 𝐸 → (∪ 𝑆 ∖ 𝑦) = (∪ 𝑆 ∖ 𝐸))
2	1	eleq1d 2817	. 2 ⊢ (𝑦 = 𝐸 → ((∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝐸) ∈ 𝑆))
3		issal 44708	. . . . 5 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
4	3	ibi 266	. . . 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 481	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
7		simpr 485	. 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → 𝐸 ∈ 𝑆)
8	2, 6, 7	rspcdva 3596	1 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060   ∖ cdif 3925  ∅c0 4302  𝒫 cpw 4580  ∪ cuni 4885   class class class wbr 5125  ωcom 7822   ≼ cdom 8903  SAlgcsalg 44702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3419  df-v 3461  df-dif 3931  df-in 3935  df-ss 3945  df-pw 4582  df-uni 4886  df-salg 44703

This theorem is referenced by:  salincl  44718  saluni  44719  saliinclf  44720  saldifcl2  44722  intsal  44724  saldifcld  44741