Step |
Hyp |
Ref |
Expression |
1 |
|
xrltnr |
⊢ ( 𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴 ) |
2 |
1
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵 ) → ¬ 𝐴 < 𝐴 ) |
3 |
|
breq2 |
⊢ ( 𝐴 = 𝐵 → ( 𝐴 < 𝐴 ↔ 𝐴 < 𝐵 ) ) |
4 |
3
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵 ) → ( 𝐴 < 𝐴 ↔ 𝐴 < 𝐵 ) ) |
5 |
2 4
|
mtbid |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵 ) → ¬ 𝐴 < 𝐵 ) |
6 |
5
|
ex |
⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 = 𝐵 → ¬ 𝐴 < 𝐵 ) ) |
7 |
6
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 = 𝐵 → ¬ 𝐴 < 𝐵 ) ) |
8 |
|
xrltnsym |
⊢ ( ( 𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( 𝐵 < 𝐴 → ¬ 𝐴 < 𝐵 ) ) |
9 |
8
|
ancoms |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐵 < 𝐴 → ¬ 𝐴 < 𝐵 ) ) |
10 |
7 9
|
jaod |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) → ¬ 𝐴 < 𝐵 ) ) |
11 |
|
elxr |
⊢ ( 𝐴 ∈ ℝ* ↔ ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞ ) ) |
12 |
|
elxr |
⊢ ( 𝐵 ∈ ℝ* ↔ ( 𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞ ) ) |
13 |
|
axlttri |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
14 |
13
|
biimprd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) → 𝐴 < 𝐵 ) ) |
15 |
14
|
con1d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
16 |
|
ltpnf |
⊢ ( 𝐴 ∈ ℝ → 𝐴 < +∞ ) |
17 |
16
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 = +∞ ) → 𝐴 < +∞ ) |
18 |
|
breq2 |
⊢ ( 𝐵 = +∞ → ( 𝐴 < 𝐵 ↔ 𝐴 < +∞ ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 = +∞ ) → ( 𝐴 < 𝐵 ↔ 𝐴 < +∞ ) ) |
20 |
17 19
|
mpbird |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 = +∞ ) → 𝐴 < 𝐵 ) |
21 |
20
|
pm2.24d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 = +∞ ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
22 |
|
mnflt |
⊢ ( 𝐴 ∈ ℝ → -∞ < 𝐴 ) |
23 |
22
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 = -∞ ) → -∞ < 𝐴 ) |
24 |
|
breq1 |
⊢ ( 𝐵 = -∞ → ( 𝐵 < 𝐴 ↔ -∞ < 𝐴 ) ) |
25 |
24
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 = -∞ ) → ( 𝐵 < 𝐴 ↔ -∞ < 𝐴 ) ) |
26 |
23 25
|
mpbird |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 = -∞ ) → 𝐵 < 𝐴 ) |
27 |
26
|
olcd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 = -∞ ) → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) |
28 |
27
|
a1d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 = -∞ ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
29 |
15 21 28
|
3jaodan |
⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞ ) ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
30 |
|
ltpnf |
⊢ ( 𝐵 ∈ ℝ → 𝐵 < +∞ ) |
31 |
30
|
adantl |
⊢ ( ( 𝐴 = +∞ ∧ 𝐵 ∈ ℝ ) → 𝐵 < +∞ ) |
32 |
|
breq2 |
⊢ ( 𝐴 = +∞ → ( 𝐵 < 𝐴 ↔ 𝐵 < +∞ ) ) |
33 |
32
|
adantr |
⊢ ( ( 𝐴 = +∞ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < 𝐴 ↔ 𝐵 < +∞ ) ) |
34 |
31 33
|
mpbird |
⊢ ( ( 𝐴 = +∞ ∧ 𝐵 ∈ ℝ ) → 𝐵 < 𝐴 ) |
35 |
34
|
olcd |
⊢ ( ( 𝐴 = +∞ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) |
36 |
35
|
a1d |
⊢ ( ( 𝐴 = +∞ ∧ 𝐵 ∈ ℝ ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
37 |
|
eqtr3 |
⊢ ( ( 𝐴 = +∞ ∧ 𝐵 = +∞ ) → 𝐴 = 𝐵 ) |
38 |
37
|
orcd |
⊢ ( ( 𝐴 = +∞ ∧ 𝐵 = +∞ ) → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) |
39 |
38
|
a1d |
⊢ ( ( 𝐴 = +∞ ∧ 𝐵 = +∞ ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
40 |
|
mnfltpnf |
⊢ -∞ < +∞ |
41 |
|
breq12 |
⊢ ( ( 𝐵 = -∞ ∧ 𝐴 = +∞ ) → ( 𝐵 < 𝐴 ↔ -∞ < +∞ ) ) |
42 |
40 41
|
mpbiri |
⊢ ( ( 𝐵 = -∞ ∧ 𝐴 = +∞ ) → 𝐵 < 𝐴 ) |
43 |
42
|
ancoms |
⊢ ( ( 𝐴 = +∞ ∧ 𝐵 = -∞ ) → 𝐵 < 𝐴 ) |
44 |
43
|
olcd |
⊢ ( ( 𝐴 = +∞ ∧ 𝐵 = -∞ ) → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) |
45 |
44
|
a1d |
⊢ ( ( 𝐴 = +∞ ∧ 𝐵 = -∞ ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
46 |
36 39 45
|
3jaodan |
⊢ ( ( 𝐴 = +∞ ∧ ( 𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞ ) ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
47 |
|
mnflt |
⊢ ( 𝐵 ∈ ℝ → -∞ < 𝐵 ) |
48 |
47
|
adantl |
⊢ ( ( 𝐴 = -∞ ∧ 𝐵 ∈ ℝ ) → -∞ < 𝐵 ) |
49 |
|
breq1 |
⊢ ( 𝐴 = -∞ → ( 𝐴 < 𝐵 ↔ -∞ < 𝐵 ) ) |
50 |
49
|
adantr |
⊢ ( ( 𝐴 = -∞ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 < 𝐵 ↔ -∞ < 𝐵 ) ) |
51 |
48 50
|
mpbird |
⊢ ( ( 𝐴 = -∞ ∧ 𝐵 ∈ ℝ ) → 𝐴 < 𝐵 ) |
52 |
51
|
pm2.24d |
⊢ ( ( 𝐴 = -∞ ∧ 𝐵 ∈ ℝ ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
53 |
|
breq12 |
⊢ ( ( 𝐴 = -∞ ∧ 𝐵 = +∞ ) → ( 𝐴 < 𝐵 ↔ -∞ < +∞ ) ) |
54 |
40 53
|
mpbiri |
⊢ ( ( 𝐴 = -∞ ∧ 𝐵 = +∞ ) → 𝐴 < 𝐵 ) |
55 |
54
|
pm2.24d |
⊢ ( ( 𝐴 = -∞ ∧ 𝐵 = +∞ ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
56 |
|
eqtr3 |
⊢ ( ( 𝐴 = -∞ ∧ 𝐵 = -∞ ) → 𝐴 = 𝐵 ) |
57 |
56
|
orcd |
⊢ ( ( 𝐴 = -∞ ∧ 𝐵 = -∞ ) → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) |
58 |
57
|
a1d |
⊢ ( ( 𝐴 = -∞ ∧ 𝐵 = -∞ ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
59 |
52 55 58
|
3jaodan |
⊢ ( ( 𝐴 = -∞ ∧ ( 𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞ ) ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
60 |
29 46 59
|
3jaoian |
⊢ ( ( ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞ ) ∧ ( 𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞ ) ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
61 |
11 12 60
|
syl2anb |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ¬ 𝐴 < 𝐵 → ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |
62 |
10 61
|
impbid |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ↔ ¬ 𝐴 < 𝐵 ) ) |
63 |
62
|
con2bid |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( 𝐴 < 𝐵 ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐵 < 𝐴 ) ) ) |