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