Theorem salunicl 43747

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 5073	. . . 4 ⊢ (𝑦 = 𝑇 → (𝑦 ≼ ω ↔ 𝑇 ≼ ω))
3		unieq 4847	. . . . 5 ⊢ (𝑦 = 𝑇 → ∪ 𝑦 = ∪ 𝑇)
4	3	eleq1d 2823	. . . 4 ⊢ (𝑦 = 𝑇 → (∪ 𝑦 ∈ 𝑆 ↔ ∪ 𝑇 ∈ 𝑆))
5	2, 4	imbi12d 344	. . 3 ⊢ (𝑦 = 𝑇 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆) ↔ (𝑇 ≼ ω → ∪ 𝑇 ∈ 𝑆)))
6		salunicl.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg)
7		issal 43745	. . . . . 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 1142	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))
11		salunicl.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑆)
12	5, 10, 11	rspcdva 3554	. 2 ⊢ (𝜑 → (𝑇 ≼ ω → ∪ 𝑇 ∈ 𝑆))
13	1, 12	mpd 15	1 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ∖ cdif 3880  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070  ωcom 7687   ≼ cdom 8689  SAlgcsalg 43739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-salg 43740

This theorem is referenced by:  saliuncl  43753  intsal  43759  smfpimbor1lem1  44219