Metamath Proof Explorer


Theorem issalnnd

Description: Sufficient condition to prove that S is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Mar-2021)

Ref Expression
Hypotheses issalnnd.s ( 𝜑𝑆𝑉 )
issalnnd.z ( 𝜑 → ∅ ∈ 𝑆 )
issalnnd.x 𝑋 = 𝑆
issalnnd.d ( ( 𝜑𝑦𝑆 ) → ( 𝑋𝑦 ) ∈ 𝑆 )
issalnnd.i ( ( 𝜑𝑒 : ℕ ⟶ 𝑆 ) → 𝑛 ∈ ℕ ( 𝑒𝑛 ) ∈ 𝑆 )
Assertion issalnnd ( 𝜑𝑆 ∈ SAlg )

Proof

Step Hyp Ref Expression
1 issalnnd.s ( 𝜑𝑆𝑉 )
2 issalnnd.z ( 𝜑 → ∅ ∈ 𝑆 )
3 issalnnd.x 𝑋 = 𝑆
4 issalnnd.d ( ( 𝜑𝑦𝑆 ) → ( 𝑋𝑦 ) ∈ 𝑆 )
5 issalnnd.i ( ( 𝜑𝑒 : ℕ ⟶ 𝑆 ) → 𝑛 ∈ ℕ ( 𝑒𝑛 ) ∈ 𝑆 )
6 unieq ( 𝑦 = ∅ → 𝑦 = ∅ )
7 uni0 ∅ = ∅
8 7 a1i ( 𝑦 = ∅ → ∅ = ∅ )
9 6 8 eqtrd ( 𝑦 = ∅ → 𝑦 = ∅ )
10 9 adantl ( ( 𝜑𝑦 = ∅ ) → 𝑦 = ∅ )
11 2 adantr ( ( 𝜑𝑦 = ∅ ) → ∅ ∈ 𝑆 )
12 10 11 eqeltrd ( ( 𝜑𝑦 = ∅ ) → 𝑦𝑆 )
13 12 3ad2antl1 ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω ) ∧ 𝑦 = ∅ ) → 𝑦𝑆 )
14 neqne ( ¬ 𝑦 = ∅ → 𝑦 ≠ ∅ )
15 14 adantl ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω ) ∧ ¬ 𝑦 = ∅ ) → 𝑦 ≠ ∅ )
16 nnfoctb ( ( 𝑦 ≼ ω ∧ 𝑦 ≠ ∅ ) → ∃ 𝑒 𝑒 : ℕ –onto𝑦 )
17 16 3ad2antl3 ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω ) ∧ 𝑦 ≠ ∅ ) → ∃ 𝑒 𝑒 : ℕ –onto𝑦 )
18 founiiun ( 𝑒 : ℕ –onto𝑦 𝑦 = 𝑛 ∈ ℕ ( 𝑒𝑛 ) )
19 18 adantl ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑒 : ℕ –onto𝑦 ) → 𝑦 = 𝑛 ∈ ℕ ( 𝑒𝑛 ) )
20 simpll ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑒 : ℕ –onto𝑦 ) → 𝜑 )
21 fof ( 𝑒 : ℕ –onto𝑦𝑒 : ℕ ⟶ 𝑦 )
22 21 adantl ( ( 𝑦 ∈ 𝒫 𝑆𝑒 : ℕ –onto𝑦 ) → 𝑒 : ℕ ⟶ 𝑦 )
23 elpwi ( 𝑦 ∈ 𝒫 𝑆𝑦𝑆 )
24 23 adantr ( ( 𝑦 ∈ 𝒫 𝑆𝑒 : ℕ –onto𝑦 ) → 𝑦𝑆 )
25 22 24 fssd ( ( 𝑦 ∈ 𝒫 𝑆𝑒 : ℕ –onto𝑦 ) → 𝑒 : ℕ ⟶ 𝑆 )
26 25 adantll ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑒 : ℕ –onto𝑦 ) → 𝑒 : ℕ ⟶ 𝑆 )
27 20 26 5 syl2anc ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑒 : ℕ –onto𝑦 ) → 𝑛 ∈ ℕ ( 𝑒𝑛 ) ∈ 𝑆 )
28 19 27 eqeltrd ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑒 : ℕ –onto𝑦 ) → 𝑦𝑆 )
29 28 ex ( ( 𝜑𝑦 ∈ 𝒫 𝑆 ) → ( 𝑒 : ℕ –onto𝑦 𝑦𝑆 ) )
30 29 adantr ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆 ) ∧ 𝑦 ≠ ∅ ) → ( 𝑒 : ℕ –onto𝑦 𝑦𝑆 ) )
31 30 3adantl3 ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω ) ∧ 𝑦 ≠ ∅ ) → ( 𝑒 : ℕ –onto𝑦 𝑦𝑆 ) )
32 31 exlimdv ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω ) ∧ 𝑦 ≠ ∅ ) → ( ∃ 𝑒 𝑒 : ℕ –onto𝑦 𝑦𝑆 ) )
33 17 32 mpd ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω ) ∧ 𝑦 ≠ ∅ ) → 𝑦𝑆 )
34 15 33 syldan ( ( ( 𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω ) ∧ ¬ 𝑦 = ∅ ) → 𝑦𝑆 )
35 13 34 pm2.61dan ( ( 𝜑𝑦 ∈ 𝒫 𝑆𝑦 ≼ ω ) → 𝑦𝑆 )
36 1 2 3 4 35 issald ( 𝜑𝑆 ∈ SAlg )