| Step | Hyp | Ref
| Expression |
| 1 | | lpfval.1 |
. . . . . . . . . . . . 13
⊢ 𝑋 = ∪
𝐽 |
| 2 | 1 | neindisj 23125 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑥 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝑥}))) → (𝑛 ∩ 𝑆) ≠ ∅) |
| 3 | 2 | expr 456 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ 𝑆) ≠ ∅)) |
| 4 | 3 | adantr 480 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ 𝑆) ≠ ∅)) |
| 5 | | difsn 4798 |
. . . . . . . . . . . . 13
⊢ (¬
𝑥 ∈ 𝑆 → (𝑆 ∖ {𝑥}) = 𝑆) |
| 6 | 5 | ineq2d 4220 |
. . . . . . . . . . . 12
⊢ (¬
𝑥 ∈ 𝑆 → (𝑛 ∩ (𝑆 ∖ {𝑥})) = (𝑛 ∩ 𝑆)) |
| 7 | 6 | neeq1d 3000 |
. . . . . . . . . . 11
⊢ (¬
𝑥 ∈ 𝑆 → ((𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅ ↔ (𝑛 ∩ 𝑆) ≠ ∅)) |
| 8 | 7 | adantl 481 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → ((𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅ ↔ (𝑛 ∩ 𝑆) ≠ ∅)) |
| 9 | 4, 8 | sylibrd 259 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) ∧ ¬ 𝑥 ∈ 𝑆) → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅)) |
| 10 | 9 | ex 412 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → (𝑛 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅))) |
| 11 | 10 | ralrimdv 3152 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅)) |
| 12 | | simpll 767 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top) |
| 13 | | simplr 769 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋) |
| 14 | 1 | clsss3 23067 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ 𝑋) |
| 15 | 14 | sselda 3983 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑥 ∈ 𝑋) |
| 16 | 1 | islp2 23153 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅)) |
| 17 | 12, 13, 15, 16 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑥})(𝑛 ∩ (𝑆 ∖ {𝑥})) ≠ ∅)) |
| 18 | 11, 17 | sylibrd 259 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 𝑥 ∈ 𝑆 → 𝑥 ∈ ((limPt‘𝐽)‘𝑆))) |
| 19 | 18 | orrd 864 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ 𝑆 ∨ 𝑥 ∈ ((limPt‘𝐽)‘𝑆))) |
| 20 | | elun 4153 |
. . . . 5
⊢ (𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ↔ (𝑥 ∈ 𝑆 ∨ 𝑥 ∈ ((limPt‘𝐽)‘𝑆))) |
| 21 | 19, 20 | sylibr 234 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝑆)) → 𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))) |
| 22 | 21 | ex 412 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝑆) → 𝑥 ∈ (𝑆 ∪ ((limPt‘𝐽)‘𝑆)))) |
| 23 | 22 | ssrdv 3989 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) ⊆ (𝑆 ∪ ((limPt‘𝐽)‘𝑆))) |
| 24 | 1 | sscls 23064 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆)) |
| 25 | 1 | lpsscls 23149 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((limPt‘𝐽)‘𝑆) ⊆ ((cls‘𝐽)‘𝑆)) |
| 26 | 24, 25 | unssd 4192 |
. 2
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∪ ((limPt‘𝐽)‘𝑆)) ⊆ ((cls‘𝐽)‘𝑆)) |
| 27 | 23, 26 | eqssd 4001 |
1
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑆 ∪ ((limPt‘𝐽)‘𝑆))) |