Step |
Hyp |
Ref |
Expression |
1 |
|
mnfnepnf |
⊢ -∞ ≠ +∞ |
2 |
|
eqeq1 |
⊢ ( 𝐴 = -∞ → ( 𝐴 = +∞ ↔ -∞ = +∞ ) ) |
3 |
2
|
necon3bbid |
⊢ ( 𝐴 = -∞ → ( ¬ 𝐴 = +∞ ↔ -∞ ≠ +∞ ) ) |
4 |
1 3
|
mpbiri |
⊢ ( 𝐴 = -∞ → ¬ 𝐴 = +∞ ) |
5 |
4
|
con2i |
⊢ ( 𝐴 = +∞ → ¬ 𝐴 = -∞ ) |
6 |
5
|
adantl |
⊢ ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) → ¬ 𝐴 = -∞ ) |
7 |
|
0xr |
⊢ 0 ∈ ℝ* |
8 |
|
nltmnf |
⊢ ( 0 ∈ ℝ* → ¬ 0 < -∞ ) |
9 |
7 8
|
ax-mp |
⊢ ¬ 0 < -∞ |
10 |
|
breq2 |
⊢ ( 𝐴 = -∞ → ( 0 < 𝐴 ↔ 0 < -∞ ) ) |
11 |
9 10
|
mtbiri |
⊢ ( 𝐴 = -∞ → ¬ 0 < 𝐴 ) |
12 |
11
|
con2i |
⊢ ( 0 < 𝐴 → ¬ 𝐴 = -∞ ) |
13 |
12
|
adantr |
⊢ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) → ¬ 𝐴 = -∞ ) |
14 |
6 13
|
jaoi |
⊢ ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) → ¬ 𝐴 = -∞ ) |
15 |
14
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) → ¬ 𝐴 = -∞ ) ) |
16 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → 𝐵 ∈ ℝ* ) |
17 |
|
xrltnsym |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( 𝐵 < 0 → ¬ 0 < 𝐵 ) ) |
18 |
16 7 17
|
sylancl |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 < 0 → ¬ 0 < 𝐵 ) ) |
19 |
18
|
adantrd |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) → ¬ 0 < 𝐵 ) ) |
20 |
|
breq2 |
⊢ ( 𝐵 = -∞ → ( 0 < 𝐵 ↔ 0 < -∞ ) ) |
21 |
9 20
|
mtbiri |
⊢ ( 𝐵 = -∞ → ¬ 0 < 𝐵 ) |
22 |
21
|
adantl |
⊢ ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 0 < 𝐵 ) |
23 |
22
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 0 < 𝐵 ) ) |
24 |
19 23
|
jaod |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) → ¬ 0 < 𝐵 ) ) |
25 |
15 24
|
orim12d |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵 ) ) ) |
26 |
|
ianor |
⊢ ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ↔ ( ¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞ ) ) |
27 |
|
orcom |
⊢ ( ( ¬ 0 < 𝐵 ∨ ¬ 𝐴 = -∞ ) ↔ ( ¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵 ) ) |
28 |
26 27
|
bitri |
⊢ ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ↔ ( ¬ 𝐴 = -∞ ∨ ¬ 0 < 𝐵 ) ) |
29 |
25 28
|
syl6ibr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ) ) |
30 |
18
|
con2d |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 0 < 𝐵 → ¬ 𝐵 < 0 ) ) |
31 |
30
|
adantrd |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) → ¬ 𝐵 < 0 ) ) |
32 |
|
pnfnlt |
⊢ ( 0 ∈ ℝ* → ¬ +∞ < 0 ) |
33 |
7 32
|
ax-mp |
⊢ ¬ +∞ < 0 |
34 |
|
simpr |
⊢ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) → 𝐵 = +∞ ) |
35 |
34
|
breq1d |
⊢ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) → ( 𝐵 < 0 ↔ +∞ < 0 ) ) |
36 |
33 35
|
mtbiri |
⊢ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) → ¬ 𝐵 < 0 ) |
37 |
36
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) → ¬ 𝐵 < 0 ) ) |
38 |
31 37
|
jaod |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) → ¬ 𝐵 < 0 ) ) |
39 |
4
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 = -∞ → ¬ 𝐴 = +∞ ) ) |
40 |
39
|
adantld |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) → ¬ 𝐴 = +∞ ) ) |
41 |
|
breq1 |
⊢ ( 𝐴 = +∞ → ( 𝐴 < 0 ↔ +∞ < 0 ) ) |
42 |
33 41
|
mtbiri |
⊢ ( 𝐴 = +∞ → ¬ 𝐴 < 0 ) |
43 |
42
|
con2i |
⊢ ( 𝐴 < 0 → ¬ 𝐴 = +∞ ) |
44 |
43
|
adantr |
⊢ ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 𝐴 = +∞ ) |
45 |
44
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 𝐴 = +∞ ) ) |
46 |
40 45
|
jaod |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) → ¬ 𝐴 = +∞ ) ) |
47 |
38 46
|
orim12d |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞ ) ) ) |
48 |
|
ianor |
⊢ ( ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ↔ ( ¬ 𝐵 < 0 ∨ ¬ 𝐴 = +∞ ) ) |
49 |
47 48
|
syl6ibr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ) |
50 |
29 49
|
jcad |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ) ) |
51 |
|
ioran |
⊢ ( ¬ ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ↔ ( ¬ ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∧ ¬ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ) |
52 |
50 51
|
syl6ibr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ) ) |
53 |
21
|
con2i |
⊢ ( 0 < 𝐵 → ¬ 𝐵 = -∞ ) |
54 |
53
|
adantr |
⊢ ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) → ¬ 𝐵 = -∞ ) |
55 |
54
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) → ¬ 𝐵 = -∞ ) ) |
56 |
|
pnfnemnf |
⊢ +∞ ≠ -∞ |
57 |
|
eqeq1 |
⊢ ( 𝐵 = +∞ → ( 𝐵 = -∞ ↔ +∞ = -∞ ) ) |
58 |
57
|
necon3bbid |
⊢ ( 𝐵 = +∞ → ( ¬ 𝐵 = -∞ ↔ +∞ ≠ -∞ ) ) |
59 |
56 58
|
mpbiri |
⊢ ( 𝐵 = +∞ → ¬ 𝐵 = -∞ ) |
60 |
59
|
adantl |
⊢ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) → ¬ 𝐵 = -∞ ) |
61 |
60
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) → ¬ 𝐵 = -∞ ) ) |
62 |
55 61
|
jaod |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) → ¬ 𝐵 = -∞ ) ) |
63 |
11
|
adantl |
⊢ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) → ¬ 0 < 𝐴 ) |
64 |
63
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) → ¬ 0 < 𝐴 ) ) |
65 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → 𝐴 ∈ ℝ* ) |
66 |
|
xrltnsym |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 0 ∈ ℝ* ) → ( 𝐴 < 0 → ¬ 0 < 𝐴 ) ) |
67 |
65 7 66
|
sylancl |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 0 → ¬ 0 < 𝐴 ) ) |
68 |
67
|
adantrd |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 0 < 𝐴 ) ) |
69 |
64 68
|
jaod |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) → ¬ 0 < 𝐴 ) ) |
70 |
62 69
|
orim12d |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴 ) ) ) |
71 |
|
ianor |
⊢ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ↔ ( ¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞ ) ) |
72 |
|
orcom |
⊢ ( ( ¬ 0 < 𝐴 ∨ ¬ 𝐵 = -∞ ) ↔ ( ¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴 ) ) |
73 |
71 72
|
bitri |
⊢ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ↔ ( ¬ 𝐵 = -∞ ∨ ¬ 0 < 𝐴 ) ) |
74 |
70 73
|
syl6ibr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ) ) |
75 |
42
|
adantl |
⊢ ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) → ¬ 𝐴 < 0 ) |
76 |
75
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) → ¬ 𝐴 < 0 ) ) |
77 |
67
|
con2d |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 0 < 𝐴 → ¬ 𝐴 < 0 ) ) |
78 |
77
|
adantrd |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) → ¬ 𝐴 < 0 ) ) |
79 |
76 78
|
jaod |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) → ¬ 𝐴 < 0 ) ) |
80 |
|
breq1 |
⊢ ( 𝐵 = +∞ → ( 𝐵 < 0 ↔ +∞ < 0 ) ) |
81 |
33 80
|
mtbiri |
⊢ ( 𝐵 = +∞ → ¬ 𝐵 < 0 ) |
82 |
81
|
con2i |
⊢ ( 𝐵 < 0 → ¬ 𝐵 = +∞ ) |
83 |
82
|
adantr |
⊢ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) → ¬ 𝐵 = +∞ ) |
84 |
59
|
con2i |
⊢ ( 𝐵 = -∞ → ¬ 𝐵 = +∞ ) |
85 |
84
|
adantl |
⊢ ( ( 𝐴 < 0 ∧ 𝐵 = -∞ ) → ¬ 𝐵 = +∞ ) |
86 |
83 85
|
jaoi |
⊢ ( ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) → ¬ 𝐵 = +∞ ) |
87 |
86
|
a1i |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) → ¬ 𝐵 = +∞ ) ) |
88 |
79 87
|
orim12d |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞ ) ) ) |
89 |
|
ianor |
⊢ ( ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ↔ ( ¬ 𝐴 < 0 ∨ ¬ 𝐵 = +∞ ) ) |
90 |
88 89
|
syl6ibr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) |
91 |
74 90
|
jcad |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) |
92 |
|
ioran |
⊢ ( ¬ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ↔ ( ¬ ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∧ ¬ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) |
93 |
91 92
|
syl6ibr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) |
94 |
52 93
|
jcad |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ( ¬ ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ¬ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) |
95 |
|
or4 |
⊢ ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ↔ ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ) ∨ ( ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) ) |
96 |
|
ioran |
⊢ ( ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ↔ ( ¬ ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∧ ¬ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) |
97 |
94 95 96
|
3imtr4g |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( ( ( 0 < 𝐵 ∧ 𝐴 = +∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = -∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = +∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = -∞ ) ) ) → ¬ ( ( ( 0 < 𝐵 ∧ 𝐴 = -∞ ) ∨ ( 𝐵 < 0 ∧ 𝐴 = +∞ ) ) ∨ ( ( 0 < 𝐴 ∧ 𝐵 = -∞ ) ∨ ( 𝐴 < 0 ∧ 𝐵 = +∞ ) ) ) ) ) |