Step |
Hyp |
Ref |
Expression |
1 |
|
elxr |
⊢ ( 𝐴 ∈ ℝ* ↔ ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞ ) ) |
2 |
|
ltnr |
⊢ ( 𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴 ) |
3 |
|
pnfnre |
⊢ +∞ ∉ ℝ |
4 |
3
|
neli |
⊢ ¬ +∞ ∈ ℝ |
5 |
4
|
intnan |
⊢ ¬ ( +∞ ∈ ℝ ∧ +∞ ∈ ℝ ) |
6 |
5
|
intnanr |
⊢ ¬ ( ( +∞ ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ +∞ <ℝ +∞ ) |
7 |
|
pnfnemnf |
⊢ +∞ ≠ -∞ |
8 |
7
|
neii |
⊢ ¬ +∞ = -∞ |
9 |
8
|
intnanr |
⊢ ¬ ( +∞ = -∞ ∧ +∞ = +∞ ) |
10 |
6 9
|
pm3.2ni |
⊢ ¬ ( ( ( +∞ ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ +∞ <ℝ +∞ ) ∨ ( +∞ = -∞ ∧ +∞ = +∞ ) ) |
11 |
4
|
intnanr |
⊢ ¬ ( +∞ ∈ ℝ ∧ +∞ = +∞ ) |
12 |
4
|
intnan |
⊢ ¬ ( +∞ = -∞ ∧ +∞ ∈ ℝ ) |
13 |
11 12
|
pm3.2ni |
⊢ ¬ ( ( +∞ ∈ ℝ ∧ +∞ = +∞ ) ∨ ( +∞ = -∞ ∧ +∞ ∈ ℝ ) ) |
14 |
10 13
|
pm3.2ni |
⊢ ¬ ( ( ( ( +∞ ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ +∞ <ℝ +∞ ) ∨ ( +∞ = -∞ ∧ +∞ = +∞ ) ) ∨ ( ( +∞ ∈ ℝ ∧ +∞ = +∞ ) ∨ ( +∞ = -∞ ∧ +∞ ∈ ℝ ) ) ) |
15 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
16 |
|
ltxr |
⊢ ( ( +∞ ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( +∞ < +∞ ↔ ( ( ( ( +∞ ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ +∞ <ℝ +∞ ) ∨ ( +∞ = -∞ ∧ +∞ = +∞ ) ) ∨ ( ( +∞ ∈ ℝ ∧ +∞ = +∞ ) ∨ ( +∞ = -∞ ∧ +∞ ∈ ℝ ) ) ) ) ) |
17 |
15 15 16
|
mp2an |
⊢ ( +∞ < +∞ ↔ ( ( ( ( +∞ ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ +∞ <ℝ +∞ ) ∨ ( +∞ = -∞ ∧ +∞ = +∞ ) ) ∨ ( ( +∞ ∈ ℝ ∧ +∞ = +∞ ) ∨ ( +∞ = -∞ ∧ +∞ ∈ ℝ ) ) ) ) |
18 |
14 17
|
mtbir |
⊢ ¬ +∞ < +∞ |
19 |
|
breq12 |
⊢ ( ( 𝐴 = +∞ ∧ 𝐴 = +∞ ) → ( 𝐴 < 𝐴 ↔ +∞ < +∞ ) ) |
20 |
19
|
anidms |
⊢ ( 𝐴 = +∞ → ( 𝐴 < 𝐴 ↔ +∞ < +∞ ) ) |
21 |
18 20
|
mtbiri |
⊢ ( 𝐴 = +∞ → ¬ 𝐴 < 𝐴 ) |
22 |
|
mnfnre |
⊢ -∞ ∉ ℝ |
23 |
22
|
neli |
⊢ ¬ -∞ ∈ ℝ |
24 |
23
|
intnan |
⊢ ¬ ( -∞ ∈ ℝ ∧ -∞ ∈ ℝ ) |
25 |
24
|
intnanr |
⊢ ¬ ( ( -∞ ∈ ℝ ∧ -∞ ∈ ℝ ) ∧ -∞ <ℝ -∞ ) |
26 |
7
|
nesymi |
⊢ ¬ -∞ = +∞ |
27 |
26
|
intnan |
⊢ ¬ ( -∞ = -∞ ∧ -∞ = +∞ ) |
28 |
25 27
|
pm3.2ni |
⊢ ¬ ( ( ( -∞ ∈ ℝ ∧ -∞ ∈ ℝ ) ∧ -∞ <ℝ -∞ ) ∨ ( -∞ = -∞ ∧ -∞ = +∞ ) ) |
29 |
23
|
intnanr |
⊢ ¬ ( -∞ ∈ ℝ ∧ -∞ = +∞ ) |
30 |
23
|
intnan |
⊢ ¬ ( -∞ = -∞ ∧ -∞ ∈ ℝ ) |
31 |
29 30
|
pm3.2ni |
⊢ ¬ ( ( -∞ ∈ ℝ ∧ -∞ = +∞ ) ∨ ( -∞ = -∞ ∧ -∞ ∈ ℝ ) ) |
32 |
28 31
|
pm3.2ni |
⊢ ¬ ( ( ( ( -∞ ∈ ℝ ∧ -∞ ∈ ℝ ) ∧ -∞ <ℝ -∞ ) ∨ ( -∞ = -∞ ∧ -∞ = +∞ ) ) ∨ ( ( -∞ ∈ ℝ ∧ -∞ = +∞ ) ∨ ( -∞ = -∞ ∧ -∞ ∈ ℝ ) ) ) |
33 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
34 |
|
ltxr |
⊢ ( ( -∞ ∈ ℝ* ∧ -∞ ∈ ℝ* ) → ( -∞ < -∞ ↔ ( ( ( ( -∞ ∈ ℝ ∧ -∞ ∈ ℝ ) ∧ -∞ <ℝ -∞ ) ∨ ( -∞ = -∞ ∧ -∞ = +∞ ) ) ∨ ( ( -∞ ∈ ℝ ∧ -∞ = +∞ ) ∨ ( -∞ = -∞ ∧ -∞ ∈ ℝ ) ) ) ) ) |
35 |
33 33 34
|
mp2an |
⊢ ( -∞ < -∞ ↔ ( ( ( ( -∞ ∈ ℝ ∧ -∞ ∈ ℝ ) ∧ -∞ <ℝ -∞ ) ∨ ( -∞ = -∞ ∧ -∞ = +∞ ) ) ∨ ( ( -∞ ∈ ℝ ∧ -∞ = +∞ ) ∨ ( -∞ = -∞ ∧ -∞ ∈ ℝ ) ) ) ) |
36 |
32 35
|
mtbir |
⊢ ¬ -∞ < -∞ |
37 |
|
breq12 |
⊢ ( ( 𝐴 = -∞ ∧ 𝐴 = -∞ ) → ( 𝐴 < 𝐴 ↔ -∞ < -∞ ) ) |
38 |
37
|
anidms |
⊢ ( 𝐴 = -∞ → ( 𝐴 < 𝐴 ↔ -∞ < -∞ ) ) |
39 |
36 38
|
mtbiri |
⊢ ( 𝐴 = -∞ → ¬ 𝐴 < 𝐴 ) |
40 |
2 21 39
|
3jaoi |
⊢ ( ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞ ) → ¬ 𝐴 < 𝐴 ) |
41 |
1 40
|
sylbi |
⊢ ( 𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴 ) |