Metamath Proof Explorer


Theorem salunicl

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

Ref Expression
Hypotheses salunicl.s ( 𝜑𝑆 ∈ SAlg )
salunicl.t ( 𝜑𝑇 ∈ 𝒫 𝑆 )
salunicl.tct ( 𝜑𝑇 ≼ ω )
Assertion salunicl ( 𝜑 𝑇𝑆 )

Proof

Step Hyp Ref Expression
1 salunicl.s ( 𝜑𝑆 ∈ SAlg )
2 salunicl.t ( 𝜑𝑇 ∈ 𝒫 𝑆 )
3 salunicl.tct ( 𝜑𝑇 ≼ ω )
4 breq1 ( 𝑦 = 𝑇 → ( 𝑦 ≼ ω ↔ 𝑇 ≼ ω ) )
5 unieq ( 𝑦 = 𝑇 𝑦 = 𝑇 )
6 5 eleq1d ( 𝑦 = 𝑇 → ( 𝑦𝑆 𝑇𝑆 ) )
7 4 6 imbi12d ( 𝑦 = 𝑇 → ( ( 𝑦 ≼ ω → 𝑦𝑆 ) ↔ ( 𝑇 ≼ ω → 𝑇𝑆 ) ) )
8 issal ( 𝑆 ∈ SAlg → ( 𝑆 ∈ SAlg ↔ ( ∅ ∈ 𝑆 ∧ ∀ 𝑦𝑆 ( 𝑆𝑦 ) ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( 𝑦 ≼ ω → 𝑦𝑆 ) ) ) )
9 1 8 syl ( 𝜑 → ( 𝑆 ∈ SAlg ↔ ( ∅ ∈ 𝑆 ∧ ∀ 𝑦𝑆 ( 𝑆𝑦 ) ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( 𝑦 ≼ ω → 𝑦𝑆 ) ) ) )
10 1 9 mpbid ( 𝜑 → ( ∅ ∈ 𝑆 ∧ ∀ 𝑦𝑆 ( 𝑆𝑦 ) ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( 𝑦 ≼ ω → 𝑦𝑆 ) ) )
11 10 simp3d ( 𝜑 → ∀ 𝑦 ∈ 𝒫 𝑆 ( 𝑦 ≼ ω → 𝑦𝑆 ) )
12 7 11 2 rspcdva ( 𝜑 → ( 𝑇 ≼ ω → 𝑇𝑆 ) )
13 3 12 mpd ( 𝜑 𝑇𝑆 )