Step |
Hyp |
Ref |
Expression |
1 |
|
reconnlem1 |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( ( topGen ‘ ran (,) ) ↾t 𝐴 ) ∈ Conn ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) |
2 |
1
|
ralrimivva |
⊢ ( ( 𝐴 ⊆ ℝ ∧ ( ( topGen ‘ ran (,) ) ↾t 𝐴 ) ∈ Conn ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) |
3 |
2
|
ex |
⊢ ( 𝐴 ⊆ ℝ → ( ( ( topGen ‘ ran (,) ) ↾t 𝐴 ) ∈ Conn → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ) |
4 |
|
n0 |
⊢ ( ( 𝑢 ∩ 𝐴 ) ≠ ∅ ↔ ∃ 𝑏 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ) |
5 |
|
n0 |
⊢ ( ( 𝑣 ∩ 𝐴 ) ≠ ∅ ↔ ∃ 𝑐 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) |
6 |
4 5
|
anbi12i |
⊢ ( ( ( 𝑢 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑣 ∩ 𝐴 ) ≠ ∅ ) ↔ ( ∃ 𝑏 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ ∃ 𝑐 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ) |
7 |
|
exdistrv |
⊢ ( ∃ 𝑏 ∃ 𝑐 ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ↔ ( ∃ 𝑏 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ ∃ 𝑐 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ) |
8 |
|
simplll |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) → 𝐴 ⊆ ℝ ) |
9 |
|
simprll |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) → 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ) |
10 |
9
|
elin2d |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) → 𝑏 ∈ 𝐴 ) |
11 |
8 10
|
sseldd |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) → 𝑏 ∈ ℝ ) |
12 |
|
simprlr |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) → 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) |
13 |
12
|
elin2d |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) → 𝑐 ∈ 𝐴 ) |
14 |
8 13
|
sseldd |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) → 𝑐 ∈ ℝ ) |
15 |
8
|
adantr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑏 ≤ 𝑐 ) → 𝐴 ⊆ ℝ ) |
16 |
|
simplrl |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) → 𝑢 ∈ ( topGen ‘ ran (,) ) ) |
17 |
16
|
ad2antrr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑏 ≤ 𝑐 ) → 𝑢 ∈ ( topGen ‘ ran (,) ) ) |
18 |
|
simplrr |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) → 𝑣 ∈ ( topGen ‘ ran (,) ) ) |
19 |
18
|
ad2antrr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑏 ≤ 𝑐 ) → 𝑣 ∈ ( topGen ‘ ran (,) ) ) |
20 |
|
simpllr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑏 ≤ 𝑐 ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) |
21 |
9
|
adantr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑏 ≤ 𝑐 ) → 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ) |
22 |
12
|
adantr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑏 ≤ 𝑐 ) → 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) |
23 |
|
simplrr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑏 ≤ 𝑐 ) → ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) |
24 |
|
simpr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑏 ≤ 𝑐 ) → 𝑏 ≤ 𝑐 ) |
25 |
|
eqid |
⊢ sup ( ( 𝑢 ∩ ( 𝑏 [,] 𝑐 ) ) , ℝ , < ) = sup ( ( 𝑢 ∩ ( 𝑏 [,] 𝑐 ) ) , ℝ , < ) |
26 |
15 17 19 20 21 22 23 24 25
|
reconnlem2 |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑏 ≤ 𝑐 ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) |
27 |
8
|
adantr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑐 ≤ 𝑏 ) → 𝐴 ⊆ ℝ ) |
28 |
18
|
ad2antrr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑐 ≤ 𝑏 ) → 𝑣 ∈ ( topGen ‘ ran (,) ) ) |
29 |
16
|
ad2antrr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑐 ≤ 𝑏 ) → 𝑢 ∈ ( topGen ‘ ran (,) ) ) |
30 |
|
simpllr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑐 ≤ 𝑏 ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) |
31 |
12
|
adantr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑐 ≤ 𝑏 ) → 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) |
32 |
9
|
adantr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑐 ≤ 𝑏 ) → 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ) |
33 |
|
incom |
⊢ ( 𝑣 ∩ 𝑢 ) = ( 𝑢 ∩ 𝑣 ) |
34 |
|
simplrr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑐 ≤ 𝑏 ) → ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) |
35 |
33 34
|
eqsstrid |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑐 ≤ 𝑏 ) → ( 𝑣 ∩ 𝑢 ) ⊆ ( ℝ ∖ 𝐴 ) ) |
36 |
|
simpr |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑐 ≤ 𝑏 ) → 𝑐 ≤ 𝑏 ) |
37 |
|
eqid |
⊢ sup ( ( 𝑣 ∩ ( 𝑐 [,] 𝑏 ) ) , ℝ , < ) = sup ( ( 𝑣 ∩ ( 𝑐 [,] 𝑏 ) ) , ℝ , < ) |
38 |
27 28 29 30 31 32 35 36 37
|
reconnlem2 |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑐 ≤ 𝑏 ) → ¬ 𝐴 ⊆ ( 𝑣 ∪ 𝑢 ) ) |
39 |
|
uncom |
⊢ ( 𝑣 ∪ 𝑢 ) = ( 𝑢 ∪ 𝑣 ) |
40 |
39
|
sseq2i |
⊢ ( 𝐴 ⊆ ( 𝑣 ∪ 𝑢 ) ↔ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) |
41 |
38 40
|
sylnib |
⊢ ( ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) ∧ 𝑐 ≤ 𝑏 ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) |
42 |
11 14 26 41
|
lecasei |
⊢ ( ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ∧ ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) |
43 |
42
|
exp32 |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) → ( ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) → ( ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) ) ) |
44 |
43
|
exlimdvv |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) → ( ∃ 𝑏 ∃ 𝑐 ( 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) → ( ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) ) ) |
45 |
7 44
|
syl5bir |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) → ( ( ∃ 𝑏 𝑏 ∈ ( 𝑢 ∩ 𝐴 ) ∧ ∃ 𝑐 𝑐 ∈ ( 𝑣 ∩ 𝐴 ) ) → ( ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) ) ) |
46 |
6 45
|
syl5bi |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) → ( ( ( 𝑢 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑣 ∩ 𝐴 ) ≠ ∅ ) → ( ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) ) ) |
47 |
46
|
expd |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) → ( ( 𝑢 ∩ 𝐴 ) ≠ ∅ → ( ( 𝑣 ∩ 𝐴 ) ≠ ∅ → ( ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) ) ) ) |
48 |
47
|
3impd |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) → ( ( ( 𝑢 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑣 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) ) |
49 |
48
|
ex |
⊢ ( ( 𝐴 ⊆ ℝ ∧ ( 𝑢 ∈ ( topGen ‘ ran (,) ) ∧ 𝑣 ∈ ( topGen ‘ ran (,) ) ) ) → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 → ( ( ( 𝑢 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑣 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) ) ) |
50 |
49
|
ralrimdvva |
⊢ ( 𝐴 ⊆ ℝ → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 → ∀ 𝑢 ∈ ( topGen ‘ ran (,) ) ∀ 𝑣 ∈ ( topGen ‘ ran (,) ) ( ( ( 𝑢 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑣 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) ) ) |
51 |
|
retopon |
⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) |
52 |
|
connsub |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ 𝐴 ⊆ ℝ ) → ( ( ( topGen ‘ ran (,) ) ↾t 𝐴 ) ∈ Conn ↔ ∀ 𝑢 ∈ ( topGen ‘ ran (,) ) ∀ 𝑣 ∈ ( topGen ‘ ran (,) ) ( ( ( 𝑢 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑣 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) ) ) |
53 |
51 52
|
mpan |
⊢ ( 𝐴 ⊆ ℝ → ( ( ( topGen ‘ ran (,) ) ↾t 𝐴 ) ∈ Conn ↔ ∀ 𝑢 ∈ ( topGen ‘ ran (,) ) ∀ 𝑣 ∈ ( topGen ‘ ran (,) ) ( ( ( 𝑢 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑣 ∩ 𝐴 ) ≠ ∅ ∧ ( 𝑢 ∩ 𝑣 ) ⊆ ( ℝ ∖ 𝐴 ) ) → ¬ 𝐴 ⊆ ( 𝑢 ∪ 𝑣 ) ) ) ) |
54 |
50 53
|
sylibrd |
⊢ ( 𝐴 ⊆ ℝ → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 → ( ( topGen ‘ ran (,) ) ↾t 𝐴 ) ∈ Conn ) ) |
55 |
3 54
|
impbid |
⊢ ( 𝐴 ⊆ ℝ → ( ( ( topGen ‘ ran (,) ) ↾t 𝐴 ) ∈ Conn ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ( 𝑥 [,] 𝑦 ) ⊆ 𝐴 ) ) |