Step |
Hyp |
Ref |
Expression |
1 |
|
prunioo |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) = ( 𝐴 [,] 𝐵 ) ) |
2 |
1
|
eleq2d |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( 𝐶 ∈ ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) ↔ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
3 |
2
|
biimpar |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐶 ∈ ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) ) |
4 |
|
elun |
⊢ ( 𝐶 ∈ ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) ↔ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 ∈ { 𝐴 , 𝐵 } ) ) |
5 |
|
elprg |
⊢ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) → ( 𝐶 ∈ { 𝐴 , 𝐵 } ↔ ( 𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ) ) ) |
6 |
5
|
orbi2d |
⊢ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) → ( ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 ∈ { 𝐴 , 𝐵 } ) ↔ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ) ) ) ) |
7 |
4 6
|
syl5bb |
⊢ ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) → ( 𝐶 ∈ ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) ↔ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ) ) ) ) |
8 |
7
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐶 ∈ ( ( 𝐴 (,) 𝐵 ) ∪ { 𝐴 , 𝐵 } ) ↔ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ) ) ) ) |
9 |
3 8
|
mpbid |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ) ) ) |
10 |
|
3orass |
⊢ ( ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ) ↔ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ) ) ) |
11 |
|
3orcoma |
⊢ ( ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ) ↔ ( 𝐶 = 𝐴 ∨ 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) |
12 |
10 11
|
bitr3i |
⊢ ( ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ ( 𝐶 = 𝐴 ∨ 𝐶 = 𝐵 ) ) ↔ ( 𝐶 = 𝐴 ∨ 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) |
13 |
9 12
|
sylib |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐶 = 𝐴 ∨ 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) |
14 |
|
lbicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
15 |
14
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐴 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
16 |
|
eleq1 |
⊢ ( 𝐶 = 𝐴 → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
17 |
16
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐴 ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
18 |
15 17
|
mpbird |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐴 ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
19 |
|
ioossicc |
⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
20 |
19
|
sseli |
⊢ ( 𝐶 ∈ ( 𝐴 (,) 𝐵 ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
21 |
20
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
22 |
|
ubicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
23 |
22
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
24 |
|
eleq1 |
⊢ ( 𝐶 = 𝐵 → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
25 |
24
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
26 |
23 25
|
mpbird |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝐶 = 𝐵 ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
27 |
18 21 26
|
3jaodan |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ ( 𝐶 = 𝐴 ∨ 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) → 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ) |
28 |
13 27
|
impbida |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 = 𝐴 ∨ 𝐶 ∈ ( 𝐴 (,) 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) |