Step | Hyp | Ref
| Expression |
1 | | lpfval.1 |
. . . . . . . . . . . . 13
⊢ 𝑋 = ∪
𝐽 |
2 | 1 | neindisj 20921 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝑥}))) → (𝑛 ∩ 𝑆) ≠ ∅) |
3 | 2 | expr 643 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ 𝑆) ≠ ∅)) |
4 | 3 | adantr 481 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ 𝑆) ≠ ∅)) |
5 | | difsn 4328 |
. . . . . . . . . . . . 13
⊢ (¬
𝑥 ∈ 𝑆 → (𝑆 ∖ {𝑥}) = 𝑆) |
6 | 5 | ineq2d 3814 |
. . . . . . . . . . . 12
⊢ (¬
𝑥 ∈ 𝑆 → (𝑛 ∩ (𝑆 ∖ {𝑥})) = (𝑛 ∩ 𝑆)) |
7 | 6 | neeq1d 2853 |
. . . . . . . . . . 11
⊢ (¬
𝑥 ∈ 𝑆 → ((𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅ ↔ (𝑛 ∩ 𝑆) ≠ ∅)) |
8 | 7 | adantl 482 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → ((𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅ ↔ (𝑛 ∩ 𝑆) ≠ ∅)) |
9 | 4, 8 | sylibrd 249 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅)) |
10 | 9 | ex 450 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))) |
11 | 10 | ralrimdv 2968 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅)) |
12 | | simpll 790 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top) |
13 | | simplr 792 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋) |
14 | 1 | clsss3 20863 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋) |
15 | 14 | sselda 3603 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑥 ∈ 𝑋) |
16 | 1 | islp2 20949 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅)) |
17 | 12, 13, 15, 16 | syl3anc 1326 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅)) |
18 | 11, 17 | sylibrd 249 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → 𝑥 ∈ ((limPt‘𝐽)‘𝑆))) |
19 | 18 | orrd 393 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ 𝑆 ∨ 𝑥 ∈ ((limPt‘𝐽)‘𝑆))) |
20 | | elun 3753 |
. . . . 5
⊢ (𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (𝑥 ∈ 𝑆 ∨ 𝑥 ∈ ((limPt‘𝐽)‘𝑆))) |
21 | 19, 20 | sylibr 224 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))) |
22 | 21 | ex 450 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) → 𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))) |
23 | 22 | ssrdv 3609 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))) |
24 | 1 | sscls 20860 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) |
25 | 1 | lpsscls 20945 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝑆)) |
26 | 24, 25 | unssd 3789 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘𝑆)) |
27 | 23, 26 | eqssd 3620 |
1
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆))) |