Theorem difunielsiga 34811

Description: A sigma-algebra is closed under complement relative to its base set. This is immediate from the definition, see issiga 34790, but the library states it nowhere in this form. (Contributed by Vincent Gonzalez, 17-Aug-2026.)

Assertion

Ref	Expression
difunielsiga	⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐴) ∈ 𝑆)

Proof of Theorem difunielsiga

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isrnsigau 34805	. . . 4 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
2	1	simprd 501	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
3	2	simp2d 1161	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆)
4		difeq2 4057	. . . 4 ⊢ (𝑥 = 𝐴 → (∪ 𝑆 ∖ 𝑥) = (∪ 𝑆 ∖ 𝐴))
5	4	eleq1d 2845	. . 3 ⊢ (𝑥 = 𝐴 → ((∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ↔ (∪ 𝑆 ∖ 𝐴) ∈ 𝑆))
6	5	rspccva 3575	. 2 ⊢ ((∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐴) ∈ 𝑆)
7	3, 6	sylan 592	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐴) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∖ cdif 4045  𝒫 cpw 4556  ∪ cuni 4866   class class class wbr 5102  ran crn 5651  ωcom 7877   ≼ cdom 8974  sigAlgebracsiga 34786

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-siga 34787

This theorem is used by:  difelsiga  34813