Step |
Hyp |
Ref |
Expression |
1 |
|
xrltletr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) ) |
2 |
|
id |
⊢ ( ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) → ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) ) |
3 |
2
|
impcom |
⊢ ( ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∧ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) ) → 𝐴 < 𝐶 ) |
4 |
|
xrltnle |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐴 ) ) |
5 |
4
|
3adant2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 < 𝐶 ↔ ¬ 𝐶 ≤ 𝐴 ) ) |
6 |
5
|
biimpd |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( 𝐴 < 𝐶 → ¬ 𝐶 ≤ 𝐴 ) ) |
7 |
6
|
imp |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 < 𝐶 ) → ¬ 𝐶 ≤ 𝐴 ) |
8 |
7
|
olcd |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ 𝐴 < 𝐶 ) → ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∨ ¬ 𝐶 ≤ 𝐴 ) ) |
9 |
8
|
expcom |
⊢ ( 𝐴 < 𝐶 → ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∨ ¬ 𝐶 ≤ 𝐴 ) ) ) |
10 |
3 9
|
syl |
⊢ ( ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∧ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) ) → ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∨ ¬ 𝐶 ≤ 𝐴 ) ) ) |
11 |
10
|
ex |
⊢ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → ( ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) → ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∨ ¬ 𝐶 ≤ 𝐴 ) ) ) ) |
12 |
11
|
com23 |
⊢ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) → ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∨ ¬ 𝐶 ≤ 𝐴 ) ) ) ) |
13 |
12
|
impd |
⊢ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) ) → ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∨ ¬ 𝐶 ≤ 𝐴 ) ) ) |
14 |
|
id |
⊢ ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ) |
15 |
14
|
orcd |
⊢ ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∨ ¬ 𝐶 ≤ 𝐴 ) ) |
16 |
15
|
a1d |
⊢ ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) ) → ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∨ ¬ 𝐶 ≤ 𝐴 ) ) ) |
17 |
13 16
|
pm2.61i |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) ) → ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∨ ¬ 𝐶 ≤ 𝐴 ) ) |
18 |
|
df-3an |
⊢ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ) ↔ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∧ 𝐶 ≤ 𝐴 ) ) |
19 |
18
|
notbii |
⊢ ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ) ↔ ¬ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∧ 𝐶 ≤ 𝐴 ) ) |
20 |
|
ianor |
⊢ ( ¬ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∧ 𝐶 ≤ 𝐴 ) ↔ ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∨ ¬ 𝐶 ≤ 𝐴 ) ) |
21 |
19 20
|
bitri |
⊢ ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ) ↔ ( ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) ∨ ¬ 𝐶 ≤ 𝐴 ) ) |
22 |
17 21
|
sylibr |
⊢ ( ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) ∧ ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) ) → ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ) ) |
23 |
22
|
ex |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ( ( ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ) → 𝐴 < 𝐶 ) → ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ) ) ) |
24 |
1 23
|
mpd |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ) → ¬ ( 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐶 ∧ 𝐶 ≤ 𝐴 ) ) |