Step |
Hyp |
Ref |
Expression |
1 |
|
elicc1 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) ) |
2 |
|
simp1 |
⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) → 𝐶 ∈ ℝ* ) |
3 |
2
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) → 𝐶 ∈ ℝ* ) ) |
4 |
|
xrletr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) → 𝐴 ≤ 𝐵 ) ) |
5 |
4
|
exp5o |
⊢ ( 𝐴 ∈ ℝ* → ( 𝐶 ∈ ℝ* → ( 𝐵 ∈ ℝ* → ( 𝐴 ≤ 𝐶 → ( 𝐶 ≤ 𝐵 → 𝐴 ≤ 𝐵 ) ) ) ) ) |
6 |
5
|
com23 |
⊢ ( 𝐴 ∈ ℝ* → ( 𝐵 ∈ ℝ* → ( 𝐶 ∈ ℝ* → ( 𝐴 ≤ 𝐶 → ( 𝐶 ≤ 𝐵 → 𝐴 ≤ 𝐵 ) ) ) ) ) |
7 |
6
|
imp5q |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) → 𝐴 ≤ 𝐵 ) ) |
8 |
|
df-ne |
⊢ ( 𝐶 ≠ 𝐴 ↔ ¬ 𝐶 = 𝐴 ) |
9 |
|
xrleltne |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ) → ( 𝐴 < 𝐶 ↔ 𝐶 ≠ 𝐴 ) ) |
10 |
9
|
biimprd |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ) → ( 𝐶 ≠ 𝐴 → 𝐴 < 𝐶 ) ) |
11 |
8 10
|
syl5bir |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ) → ( ¬ 𝐶 = 𝐴 → 𝐴 < 𝐶 ) ) |
12 |
11
|
3adant3r3 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → ( ¬ 𝐶 = 𝐴 → 𝐴 < 𝐶 ) ) |
13 |
12
|
adantlr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → ( ¬ 𝐶 = 𝐴 → 𝐴 < 𝐶 ) ) |
14 |
|
eqcom |
⊢ ( 𝐶 = 𝐵 ↔ 𝐵 = 𝐶 ) |
15 |
14
|
necon3bbii |
⊢ ( ¬ 𝐶 = 𝐵 ↔ 𝐵 ≠ 𝐶 ) |
16 |
|
xrleltne |
⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ≤ 𝐵 ) → ( 𝐶 < 𝐵 ↔ 𝐵 ≠ 𝐶 ) ) |
17 |
16
|
biimprd |
⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ≤ 𝐵 ) → ( 𝐵 ≠ 𝐶 → 𝐶 < 𝐵 ) ) |
18 |
15 17
|
syl5bi |
⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ≤ 𝐵 ) → ( ¬ 𝐶 = 𝐵 → 𝐶 < 𝐵 ) ) |
19 |
18
|
3exp |
⊢ ( 𝐶 ∈ ℝ* → ( 𝐵 ∈ ℝ* → ( 𝐶 ≤ 𝐵 → ( ¬ 𝐶 = 𝐵 → 𝐶 < 𝐵 ) ) ) ) |
20 |
19
|
com12 |
⊢ ( 𝐵 ∈ ℝ* → ( 𝐶 ∈ ℝ* → ( 𝐶 ≤ 𝐵 → ( ¬ 𝐶 = 𝐵 → 𝐶 < 𝐵 ) ) ) ) |
21 |
20
|
imp32 |
⊢ ( ( 𝐵 ∈ ℝ* ∧ ( 𝐶 ∈ ℝ* ∧ 𝐶 ≤ 𝐵 ) ) → ( ¬ 𝐶 = 𝐵 → 𝐶 < 𝐵 ) ) |
22 |
21
|
3adantr2 |
⊢ ( ( 𝐵 ∈ ℝ* ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → ( ¬ 𝐶 = 𝐵 → 𝐶 < 𝐵 ) ) |
23 |
22
|
adantll |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → ( ¬ 𝐶 = 𝐵 → 𝐶 < 𝐵 ) ) |
24 |
13 23
|
anim12d |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) → ( ( ¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵 ) → ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) ) |
25 |
24
|
ex |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) → ( ( ¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵 ) → ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) ) ) |
26 |
|
df-or |
⊢ ( ( 𝐶 = 𝐴 ∨ ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ↔ ( ¬ 𝐶 = 𝐴 → ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) |
27 |
|
3orass |
⊢ ( ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ↔ ( 𝐶 = 𝐴 ∨ ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) |
28 |
|
pm5.6 |
⊢ ( ( ( ¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵 ) → ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) ↔ ( ¬ 𝐶 = 𝐴 → ( 𝐶 = 𝐵 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) ) ) |
29 |
|
orcom |
⊢ ( ( 𝐶 = 𝐵 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) ↔ ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) |
30 |
29
|
imbi2i |
⊢ ( ( ¬ 𝐶 = 𝐴 → ( 𝐶 = 𝐵 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) ) ↔ ( ¬ 𝐶 = 𝐴 → ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) |
31 |
28 30
|
bitri |
⊢ ( ( ( ¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵 ) → ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) ↔ ( ¬ 𝐶 = 𝐴 → ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) |
32 |
26 27 31
|
3bitr4ri |
⊢ ( ( ( ¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵 ) → ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ) ↔ ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) |
33 |
25 32
|
syl6ib |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) → ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) |
34 |
3 7 33
|
3jcad |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) → ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) ) |
35 |
|
simp1 |
⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) → 𝐶 ∈ ℝ* ) |
36 |
35
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) → 𝐶 ∈ ℝ* ) ) |
37 |
|
xrleid |
⊢ ( 𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴 ) |
38 |
37
|
ad3antrrr |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ 𝐴 ) |
39 |
|
breq2 |
⊢ ( 𝐶 = 𝐴 → ( 𝐴 ≤ 𝐶 ↔ 𝐴 ≤ 𝐴 ) ) |
40 |
38 39
|
syl5ibrcom |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐶 = 𝐴 → 𝐴 ≤ 𝐶 ) ) |
41 |
|
xrltle |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 < 𝐶 → 𝐴 ≤ 𝐶 ) ) |
42 |
41
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐶 → 𝐴 ≤ 𝐶 ) ) |
43 |
42
|
adantllr |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐴 < 𝐶 → 𝐴 ≤ 𝐶 ) ) |
44 |
43
|
adantrd |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) → 𝐴 ≤ 𝐶 ) ) |
45 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ≤ 𝐵 ) |
46 |
|
breq2 |
⊢ ( 𝐶 = 𝐵 → ( 𝐴 ≤ 𝐶 ↔ 𝐴 ≤ 𝐵 ) ) |
47 |
45 46
|
syl5ibrcom |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐶 = 𝐵 → 𝐴 ≤ 𝐶 ) ) |
48 |
40 44 47
|
3jaod |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) → 𝐴 ≤ 𝐶 ) ) |
49 |
48
|
exp31 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ℝ* → ( 𝐴 ≤ 𝐵 → ( ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) → 𝐴 ≤ 𝐶 ) ) ) ) |
50 |
49
|
3impd |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) → 𝐴 ≤ 𝐶 ) ) |
51 |
|
breq1 |
⊢ ( 𝐶 = 𝐴 → ( 𝐶 ≤ 𝐵 ↔ 𝐴 ≤ 𝐵 ) ) |
52 |
45 51
|
syl5ibrcom |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐶 = 𝐴 → 𝐶 ≤ 𝐵 ) ) |
53 |
|
xrltle |
⊢ ( ( 𝐶 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 < 𝐵 → 𝐶 ≤ 𝐵 ) ) |
54 |
53
|
ancoms |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐶 < 𝐵 → 𝐶 ≤ 𝐵 ) ) |
55 |
54
|
adantld |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) → 𝐶 ≤ 𝐵 ) ) |
56 |
55
|
adantll |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) → 𝐶 ≤ 𝐵 ) ) |
57 |
56
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) → 𝐶 ≤ 𝐵 ) ) |
58 |
|
xrleid |
⊢ ( 𝐵 ∈ ℝ* → 𝐵 ≤ 𝐵 ) |
59 |
58
|
ad3antlr |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ≤ 𝐵 ) |
60 |
|
breq1 |
⊢ ( 𝐶 = 𝐵 → ( 𝐶 ≤ 𝐵 ↔ 𝐵 ≤ 𝐵 ) ) |
61 |
59 60
|
syl5ibrcom |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( 𝐶 = 𝐵 → 𝐶 ≤ 𝐵 ) ) |
62 |
52 57 61
|
3jaod |
⊢ ( ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 ≤ 𝐵 ) → ( ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) → 𝐶 ≤ 𝐵 ) ) |
63 |
62
|
exp31 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ℝ* → ( 𝐴 ≤ 𝐵 → ( ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) → 𝐶 ≤ 𝐵 ) ) ) ) |
64 |
63
|
3impd |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) → 𝐶 ≤ 𝐵 ) ) |
65 |
36 50 64
|
3jcad |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) → ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ) ) |
66 |
34 65
|
impbid |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) ) |
67 |
1 66
|
bitrd |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐶 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝐶 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ∧ ( 𝐶 = 𝐴 ∨ ( 𝐴 < 𝐶 ∧ 𝐶 < 𝐵 ) ∨ 𝐶 = 𝐵 ) ) ) ) |