Step |
Hyp |
Ref |
Expression |
1 |
|
simpl1 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) → 𝐴 ∈ ℝ* ) |
2 |
|
simpl2 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) → 𝐵 ∈ ℝ* ) |
3 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
4 |
|
iccleub |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑥 ≤ 𝐵 ) |
5 |
1 2 3 4
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) → 𝑥 ≤ 𝐵 ) |
6 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) → 𝐶 ∈ ℝ* ) |
7 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) → 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) |
8 |
|
iccgelb |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) → 𝐵 ≤ 𝑥 ) |
9 |
2 6 7 8
|
syl3anc |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) → 𝐵 ≤ 𝑥 ) |
10 |
|
eliccxr |
⊢ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) → 𝑥 ∈ ℝ* ) |
11 |
3 10
|
syl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) → 𝑥 ∈ ℝ* ) |
12 |
11 2
|
jca |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) → ( 𝑥 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ) |
13 |
|
xrletri3 |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝑥 = 𝐵 ↔ ( 𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥 ) ) ) |
14 |
12 13
|
syl |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) → ( 𝑥 = 𝐵 ↔ ( 𝑥 ≤ 𝐵 ∧ 𝐵 ≤ 𝑥 ) ) ) |
15 |
5 9 14
|
mpbir2and |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) → 𝑥 = 𝐵 ) |
16 |
15
|
ex |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) → 𝑥 = 𝐵 ) ) |
17 |
16
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) → 𝑥 = 𝐵 ) ) |
18 |
|
simpll1 |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) ∧ 𝑥 = 𝐵 ) → 𝐴 ∈ ℝ* ) |
19 |
|
simpll2 |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) ∧ 𝑥 = 𝐵 ) → 𝐵 ∈ ℝ* ) |
20 |
|
simplrl |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) ∧ 𝑥 = 𝐵 ) → 𝐴 ≤ 𝐵 ) |
21 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) ∧ 𝑥 = 𝐵 ) → 𝑥 = 𝐵 ) |
22 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 = 𝐵 ) → 𝑥 = 𝐵 ) |
23 |
|
ubicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
24 |
23
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 = 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
25 |
22 24
|
eqeltrd |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) ∧ 𝑥 = 𝐵 ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
26 |
18 19 20 21 25
|
syl31anc |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) ∧ 𝑥 = 𝐵 ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
27 |
|
simpll3 |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) ∧ 𝑥 = 𝐵 ) → 𝐶 ∈ ℝ* ) |
28 |
|
simplrr |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) ∧ 𝑥 = 𝐵 ) → 𝐵 ≤ 𝐶 ) |
29 |
|
simpr |
⊢ ( ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐵 ≤ 𝐶 ) ∧ 𝑥 = 𝐵 ) → 𝑥 = 𝐵 ) |
30 |
|
lbicc2 |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐵 ≤ 𝐶 ) → 𝐵 ∈ ( 𝐵 [,] 𝐶 ) ) |
31 |
30
|
adantr |
⊢ ( ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐵 ≤ 𝐶 ) ∧ 𝑥 = 𝐵 ) → 𝐵 ∈ ( 𝐵 [,] 𝐶 ) ) |
32 |
29 31
|
eqeltrd |
⊢ ( ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐵 ≤ 𝐶 ) ∧ 𝑥 = 𝐵 ) → 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) |
33 |
19 27 28 21 32
|
syl31anc |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) ∧ 𝑥 = 𝐵 ) → 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) |
34 |
26 33
|
jca |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) ∧ 𝑥 = 𝐵 ) → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) |
35 |
34
|
ex |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) → ( 𝑥 = 𝐵 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) ) |
36 |
17 35
|
impbid |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) → ( ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ↔ 𝑥 = 𝐵 ) ) |
37 |
|
elin |
⊢ ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) ∩ ( 𝐵 [,] 𝐶 ) ) ↔ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝑥 ∈ ( 𝐵 [,] 𝐶 ) ) ) |
38 |
|
velsn |
⊢ ( 𝑥 ∈ { 𝐵 } ↔ 𝑥 = 𝐵 ) |
39 |
36 37 38
|
3bitr4g |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) → ( 𝑥 ∈ ( ( 𝐴 [,] 𝐵 ) ∩ ( 𝐵 [,] 𝐶 ) ) ↔ 𝑥 ∈ { 𝐵 } ) ) |
40 |
39
|
eqrdv |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) → ( ( 𝐴 [,] 𝐵 ) ∩ ( 𝐵 [,] 𝐶 ) ) = { 𝐵 } ) |