Step |
Hyp |
Ref |
Expression |
1 |
|
eleq2 |
⊢ ( 𝑥 = 𝑆 → ( ∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆 ) ) |
2 |
|
id |
⊢ ( 𝑥 = 𝑆 → 𝑥 = 𝑆 ) |
3 |
|
unieq |
⊢ ( 𝑥 = 𝑆 → ∪ 𝑥 = ∪ 𝑆 ) |
4 |
3
|
difeq1d |
⊢ ( 𝑥 = 𝑆 → ( ∪ 𝑥 ∖ 𝑦 ) = ( ∪ 𝑆 ∖ 𝑦 ) ) |
5 |
4 2
|
eleq12d |
⊢ ( 𝑥 = 𝑆 → ( ( ∪ 𝑥 ∖ 𝑦 ) ∈ 𝑥 ↔ ( ∪ 𝑆 ∖ 𝑦 ) ∈ 𝑆 ) ) |
6 |
2 5
|
raleqbidv |
⊢ ( 𝑥 = 𝑆 → ( ∀ 𝑦 ∈ 𝑥 ( ∪ 𝑥 ∖ 𝑦 ) ∈ 𝑥 ↔ ∀ 𝑦 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑦 ) ∈ 𝑆 ) ) |
7 |
|
pweq |
⊢ ( 𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆 ) |
8 |
|
eleq2 |
⊢ ( 𝑥 = 𝑆 → ( ∪ 𝑦 ∈ 𝑥 ↔ ∪ 𝑦 ∈ 𝑆 ) ) |
9 |
8
|
imbi2d |
⊢ ( 𝑥 = 𝑆 → ( ( 𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥 ) ↔ ( 𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆 ) ) ) |
10 |
7 9
|
raleqbidv |
⊢ ( 𝑥 = 𝑆 → ( ∀ 𝑦 ∈ 𝒫 𝑥 ( 𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ∈ 𝒫 𝑆 ( 𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆 ) ) ) |
11 |
1 6 10
|
3anbi123d |
⊢ ( 𝑥 = 𝑆 → ( ( ∅ ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( ∪ 𝑥 ∖ 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝒫 𝑥 ( 𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥 ) ) ↔ ( ∅ ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑦 ) ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( 𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆 ) ) ) ) |
12 |
|
df-salg |
⊢ SAlg = { 𝑥 ∣ ( ∅ ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( ∪ 𝑥 ∖ 𝑦 ) ∈ 𝑥 ∧ ∀ 𝑦 ∈ 𝒫 𝑥 ( 𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥 ) ) } |
13 |
11 12
|
elab2g |
⊢ ( 𝑆 ∈ 𝑉 → ( 𝑆 ∈ SAlg ↔ ( ∅ ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝑆 ( ∪ 𝑆 ∖ 𝑦 ) ∈ 𝑆 ∧ ∀ 𝑦 ∈ 𝒫 𝑆 ( 𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆 ) ) ) ) |