Step |
Hyp |
Ref |
Expression |
1 |
|
1stcclb.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
1
|
is1stc2 |
⊢ ( 𝐽 ∈ 1stω ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑤 ∈ 𝑋 ∃ 𝑥 ∈ 𝒫 𝐽 ( 𝑥 ≼ ω ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑤 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) ) ) |
3 |
2
|
simprbi |
⊢ ( 𝐽 ∈ 1stω → ∀ 𝑤 ∈ 𝑋 ∃ 𝑥 ∈ 𝒫 𝐽 ( 𝑥 ≼ ω ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑤 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) ) |
4 |
|
eleq1 |
⊢ ( 𝑤 = 𝐴 → ( 𝑤 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦 ) ) |
5 |
|
eleq1 |
⊢ ( 𝑤 = 𝐴 → ( 𝑤 ∈ 𝑧 ↔ 𝐴 ∈ 𝑧 ) ) |
6 |
5
|
anbi1d |
⊢ ( 𝑤 = 𝐴 → ( ( 𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ↔ ( 𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) |
7 |
6
|
rexbidv |
⊢ ( 𝑤 = 𝐴 → ( ∃ 𝑧 ∈ 𝑥 ( 𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ↔ ∃ 𝑧 ∈ 𝑥 ( 𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) |
8 |
4 7
|
imbi12d |
⊢ ( 𝑤 = 𝐴 → ( ( 𝑤 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ↔ ( 𝐴 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) ) |
9 |
8
|
ralbidv |
⊢ ( 𝑤 = 𝐴 → ( ∀ 𝑦 ∈ 𝐽 ( 𝑤 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ↔ ∀ 𝑦 ∈ 𝐽 ( 𝐴 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) ) |
10 |
9
|
anbi2d |
⊢ ( 𝑤 = 𝐴 → ( ( 𝑥 ≼ ω ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑤 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) ↔ ( 𝑥 ≼ ω ∧ ∀ 𝑦 ∈ 𝐽 ( 𝐴 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) ) ) |
11 |
10
|
rexbidv |
⊢ ( 𝑤 = 𝐴 → ( ∃ 𝑥 ∈ 𝒫 𝐽 ( 𝑥 ≼ ω ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑤 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) ↔ ∃ 𝑥 ∈ 𝒫 𝐽 ( 𝑥 ≼ ω ∧ ∀ 𝑦 ∈ 𝐽 ( 𝐴 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) ) ) |
12 |
11
|
rspcv |
⊢ ( 𝐴 ∈ 𝑋 → ( ∀ 𝑤 ∈ 𝑋 ∃ 𝑥 ∈ 𝒫 𝐽 ( 𝑥 ≼ ω ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑤 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝑤 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) → ∃ 𝑥 ∈ 𝒫 𝐽 ( 𝑥 ≼ ω ∧ ∀ 𝑦 ∈ 𝐽 ( 𝐴 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) ) ) |
13 |
3 12
|
mpan9 |
⊢ ( ( 𝐽 ∈ 1stω ∧ 𝐴 ∈ 𝑋 ) → ∃ 𝑥 ∈ 𝒫 𝐽 ( 𝑥 ≼ ω ∧ ∀ 𝑦 ∈ 𝐽 ( 𝐴 ∈ 𝑦 → ∃ 𝑧 ∈ 𝑥 ( 𝐴 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) ) ) |