Theorem salunicl 43857

Description: SAlg sigma-algebra is closed under countable union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
salunicl.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
salunicl.t	⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑆)
salunicl.tct	⊢ (𝜑 → 𝑇 ≼ ω)

Assertion

Ref	Expression
salunicl	⊢ (𝜑 → ∪ 𝑇 ∈ 𝑆)

Proof of Theorem salunicl

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		salunicl.tct	. 2 ⊢ (𝜑 → 𝑇 ≼ ω)
2		breq1 5077	. . . 4 ⊢ (𝑦 = 𝑇 → (𝑦 ≼ ω ↔ 𝑇 ≼ ω))
3		unieq 4850	. . . . 5 ⊢ (𝑦 = 𝑇 → ∪ 𝑦 = ∪ 𝑇)
4	3	eleq1d 2823	. . . 4 ⊢ (𝑦 = 𝑇 → (∪ 𝑦 ∈ 𝑆 ↔ ∪ 𝑇 ∈ 𝑆))
5	2, 4	imbi12d 345	. . 3 ⊢ (𝑦 = 𝑇 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆) ↔ (𝑇 ≼ ω → ∪ 𝑇 ∈ 𝑆)))
6		salunicl.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg)
7		issal 43855	. . . . . 6 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
9	6, 8	mpbid 231	. . . 4 ⊢ (𝜑 → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
10	9	simp3d 1143	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))
11		salunicl.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑆)
12	5, 10, 11	rspcdva 3562	. 2 ⊢ (𝜑 → (𝑇 ≼ ω → ∪ 𝑇 ∈ 𝑆))
13	1, 12	mpd 15	1 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ∖ cdif 3884  ∅c0 4256  𝒫 cpw 4533  ∪ cuni 4839   class class class wbr 5074  ωcom 7712   ≼ cdom 8731  SAlgcsalg 43849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-salg 43850

This theorem is referenced by:  saliuncl  43863  intsal  43869  smfpimbor1lem1  44332