Step |
Hyp |
Ref |
Expression |
1 |
|
ist0.1 |
⊢ 𝑋 = ∪ 𝐽 |
2 |
1
|
ishaus |
⊢ ( 𝐽 ∈ Haus ↔ ( 𝐽 ∈ Top ∧ ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 ≠ 𝑦 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
3 |
2
|
simprbi |
⊢ ( 𝐽 ∈ Haus → ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 ≠ 𝑦 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) |
4 |
|
neeq1 |
⊢ ( 𝑥 = 𝑃 → ( 𝑥 ≠ 𝑦 ↔ 𝑃 ≠ 𝑦 ) ) |
5 |
|
eleq1 |
⊢ ( 𝑥 = 𝑃 → ( 𝑥 ∈ 𝑛 ↔ 𝑃 ∈ 𝑛 ) ) |
6 |
5
|
3anbi1d |
⊢ ( 𝑥 = 𝑃 → ( ( 𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ↔ ( 𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) |
7 |
6
|
2rexbidv |
⊢ ( 𝑥 = 𝑃 → ( ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ↔ ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) |
8 |
4 7
|
imbi12d |
⊢ ( 𝑥 = 𝑃 → ( ( 𝑥 ≠ 𝑦 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ↔ ( 𝑃 ≠ 𝑦 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
9 |
|
neeq2 |
⊢ ( 𝑦 = 𝑄 → ( 𝑃 ≠ 𝑦 ↔ 𝑃 ≠ 𝑄 ) ) |
10 |
|
eleq1 |
⊢ ( 𝑦 = 𝑄 → ( 𝑦 ∈ 𝑚 ↔ 𝑄 ∈ 𝑚 ) ) |
11 |
10
|
3anbi2d |
⊢ ( 𝑦 = 𝑄 → ( ( 𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ↔ ( 𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) |
12 |
11
|
2rexbidv |
⊢ ( 𝑦 = 𝑄 → ( ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ↔ ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) |
13 |
9 12
|
imbi12d |
⊢ ( 𝑦 = 𝑄 → ( ( 𝑃 ≠ 𝑦 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑃 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ↔ ( 𝑃 ≠ 𝑄 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
14 |
8 13
|
rspc2v |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ) → ( ∀ 𝑥 ∈ 𝑋 ∀ 𝑦 ∈ 𝑋 ( 𝑥 ≠ 𝑦 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑥 ∈ 𝑛 ∧ 𝑦 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) → ( 𝑃 ≠ 𝑄 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
15 |
3 14
|
syl5 |
⊢ ( ( 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ) → ( 𝐽 ∈ Haus → ( 𝑃 ≠ 𝑄 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) |
16 |
15
|
ex |
⊢ ( 𝑃 ∈ 𝑋 → ( 𝑄 ∈ 𝑋 → ( 𝐽 ∈ Haus → ( 𝑃 ≠ 𝑄 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) ) |
17 |
16
|
com3r |
⊢ ( 𝐽 ∈ Haus → ( 𝑃 ∈ 𝑋 → ( 𝑄 ∈ 𝑋 → ( 𝑃 ≠ 𝑄 → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) ) ) ) |
18 |
17
|
3imp2 |
⊢ ( ( 𝐽 ∈ Haus ∧ ( 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋 ∧ 𝑃 ≠ 𝑄 ) ) → ∃ 𝑛 ∈ 𝐽 ∃ 𝑚 ∈ 𝐽 ( 𝑃 ∈ 𝑛 ∧ 𝑄 ∈ 𝑚 ∧ ( 𝑛 ∩ 𝑚 ) = ∅ ) ) |