Step |
Hyp |
Ref |
Expression |
1 |
|
infxr.x |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
infxr.y |
⊢ Ⅎ 𝑦 𝜑 |
3 |
|
infxr.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ* ) |
4 |
|
infxr.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
5 |
|
infxr.n |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵 ) |
6 |
|
infxr.e |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) |
7 |
6
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) |
8 |
7
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ 𝑥 ∈ ℝ ) → ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) |
9 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ ¬ 𝑥 ∈ ℝ ) ∧ 𝐵 < 𝑥 ) → 𝜑 ) |
10 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ ¬ 𝑥 ∈ ℝ ) ∧ 𝐵 < 𝑥 ) → 𝑥 ∈ ℝ* ) |
11 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ ¬ 𝑥 ∈ ℝ ) ∧ 𝐵 < 𝑥 ) → ¬ 𝑥 ∈ ℝ ) |
12 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
13 |
12
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ 𝐵 < 𝑥 ) → -∞ ∈ ℝ* ) |
14 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ 𝐵 < 𝑥 ) → 𝑥 ∈ ℝ* ) |
15 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ 𝐵 < 𝑥 ) → 𝐵 ∈ ℝ* ) |
16 |
|
mnfle |
⊢ ( 𝐵 ∈ ℝ* → -∞ ≤ 𝐵 ) |
17 |
4 16
|
syl |
⊢ ( 𝜑 → -∞ ≤ 𝐵 ) |
18 |
17
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ 𝐵 < 𝑥 ) → -∞ ≤ 𝐵 ) |
19 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ 𝐵 < 𝑥 ) → 𝐵 < 𝑥 ) |
20 |
13 15 14 18 19
|
xrlelttrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ 𝐵 < 𝑥 ) → -∞ < 𝑥 ) |
21 |
13 14 20
|
xrgtned |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ 𝐵 < 𝑥 ) → 𝑥 ≠ -∞ ) |
22 |
21
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ ¬ 𝑥 ∈ ℝ ) ∧ 𝐵 < 𝑥 ) → 𝑥 ≠ -∞ ) |
23 |
10 11 22
|
xrnmnfpnf |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ ¬ 𝑥 ∈ ℝ ) ∧ 𝐵 < 𝑥 ) → 𝑥 = +∞ ) |
24 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ ¬ 𝑥 ∈ ℝ ) ∧ 𝐵 < 𝑥 ) → 𝐵 < 𝑥 ) |
25 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝐵 = -∞ ) → 𝜑 ) |
26 |
|
id |
⊢ ( 𝐵 = -∞ → 𝐵 = -∞ ) |
27 |
|
1re |
⊢ 1 ∈ ℝ |
28 |
|
mnflt |
⊢ ( 1 ∈ ℝ → -∞ < 1 ) |
29 |
27 28
|
ax-mp |
⊢ -∞ < 1 |
30 |
26 29
|
eqbrtrdi |
⊢ ( 𝐵 = -∞ → 𝐵 < 1 ) |
31 |
30
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐵 = -∞ ) → 𝐵 < 1 ) |
32 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
33 |
|
breq2 |
⊢ ( 𝑥 = 1 → ( 𝐵 < 𝑥 ↔ 𝐵 < 1 ) ) |
34 |
|
breq2 |
⊢ ( 𝑥 = 1 → ( 𝑦 < 𝑥 ↔ 𝑦 < 1 ) ) |
35 |
34
|
rexbidv |
⊢ ( 𝑥 = 1 → ( ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 𝑦 < 1 ) ) |
36 |
33 35
|
imbi12d |
⊢ ( 𝑥 = 1 → ( ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ↔ ( 𝐵 < 1 → ∃ 𝑦 ∈ 𝐴 𝑦 < 1 ) ) ) |
37 |
36
|
rspcva |
⊢ ( ( 1 ∈ ℝ ∧ ∀ 𝑥 ∈ ℝ ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) → ( 𝐵 < 1 → ∃ 𝑦 ∈ 𝐴 𝑦 < 1 ) ) |
38 |
32 6 37
|
syl2anc |
⊢ ( 𝜑 → ( 𝐵 < 1 → ∃ 𝑦 ∈ 𝐴 𝑦 < 1 ) ) |
39 |
25 31 38
|
sylc |
⊢ ( ( 𝜑 ∧ 𝐵 = -∞ ) → ∃ 𝑦 ∈ 𝐴 𝑦 < 1 ) |
40 |
39
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ) ∧ 𝐵 = -∞ ) → ∃ 𝑦 ∈ 𝐴 𝑦 < 1 ) |
41 |
|
nfv |
⊢ Ⅎ 𝑦 𝑥 = +∞ |
42 |
2 41
|
nfan |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝑥 = +∞ ) |
43 |
3
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ ℝ* ) |
44 |
43
|
ad4ant13 |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < 1 ) → 𝑦 ∈ ℝ* ) |
45 |
|
1xr |
⊢ 1 ∈ ℝ* |
46 |
45
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < 1 ) → 1 ∈ ℝ* ) |
47 |
|
id |
⊢ ( 𝑥 = +∞ → 𝑥 = +∞ ) |
48 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
49 |
47 48
|
eqeltrdi |
⊢ ( 𝑥 = +∞ → 𝑥 ∈ ℝ* ) |
50 |
49
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = +∞ ) → 𝑥 ∈ ℝ* ) |
51 |
50
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < 1 ) → 𝑥 ∈ ℝ* ) |
52 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < 1 ) → 𝑦 < 1 ) |
53 |
|
ltpnf |
⊢ ( 1 ∈ ℝ → 1 < +∞ ) |
54 |
27 53
|
ax-mp |
⊢ 1 < +∞ |
55 |
54
|
a1i |
⊢ ( 𝑥 = +∞ → 1 < +∞ ) |
56 |
47
|
eqcomd |
⊢ ( 𝑥 = +∞ → +∞ = 𝑥 ) |
57 |
55 56
|
breqtrd |
⊢ ( 𝑥 = +∞ → 1 < 𝑥 ) |
58 |
57
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = +∞ ) → 1 < 𝑥 ) |
59 |
58
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < 1 ) → 1 < 𝑥 ) |
60 |
44 46 51 52 59
|
xrlttrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < 1 ) → 𝑦 < 𝑥 ) |
61 |
60
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑦 < 1 → 𝑦 < 𝑥 ) ) |
62 |
61
|
ex |
⊢ ( ( 𝜑 ∧ 𝑥 = +∞ ) → ( 𝑦 ∈ 𝐴 → ( 𝑦 < 1 → 𝑦 < 𝑥 ) ) ) |
63 |
42 62
|
reximdai |
⊢ ( ( 𝜑 ∧ 𝑥 = +∞ ) → ( ∃ 𝑦 ∈ 𝐴 𝑦 < 1 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) |
64 |
63
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ) ∧ 𝐵 = -∞ ) → ( ∃ 𝑦 ∈ 𝐴 𝑦 < 1 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) |
65 |
40 64
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ) ∧ 𝐵 = -∞ ) → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) |
66 |
65
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ 𝐵 = -∞ ) → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) |
67 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐵 = -∞ ) → 𝐵 ∈ ℝ* ) |
68 |
67
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → 𝐵 ∈ ℝ* ) |
69 |
26
|
necon3bi |
⊢ ( ¬ 𝐵 = -∞ → 𝐵 ≠ -∞ ) |
70 |
69
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → 𝐵 ≠ -∞ ) |
71 |
48
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → +∞ ∈ ℝ* ) |
72 |
|
simpr |
⊢ ( ( 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) → 𝐵 < 𝑥 ) |
73 |
|
simpl |
⊢ ( ( 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) → 𝑥 = +∞ ) |
74 |
72 73
|
breqtrd |
⊢ ( ( 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) → 𝐵 < +∞ ) |
75 |
74
|
3adant1 |
⊢ ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) → 𝐵 < +∞ ) |
76 |
75
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → 𝐵 < +∞ ) |
77 |
68 71 76
|
xrltned |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → 𝐵 ≠ +∞ ) |
78 |
68 70 77
|
xrred |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → 𝐵 ∈ ℝ ) |
79 |
27
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → 1 ∈ ℝ ) |
80 |
78 79
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → ( 𝐵 + 1 ) ∈ ℝ ) |
81 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐵 = -∞ ) → ∀ 𝑥 ∈ ℝ ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) |
82 |
81
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → ∀ 𝑥 ∈ ℝ ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) |
83 |
80 82
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → ( ( 𝐵 + 1 ) ∈ ℝ ∧ ∀ 𝑥 ∈ ℝ ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) ) |
84 |
78
|
ltp1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → 𝐵 < ( 𝐵 + 1 ) ) |
85 |
|
breq2 |
⊢ ( 𝑥 = ( 𝐵 + 1 ) → ( 𝐵 < 𝑥 ↔ 𝐵 < ( 𝐵 + 1 ) ) ) |
86 |
|
breq2 |
⊢ ( 𝑥 = ( 𝐵 + 1 ) → ( 𝑦 < 𝑥 ↔ 𝑦 < ( 𝐵 + 1 ) ) ) |
87 |
86
|
rexbidv |
⊢ ( 𝑥 = ( 𝐵 + 1 ) → ( ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 𝑦 < ( 𝐵 + 1 ) ) ) |
88 |
85 87
|
imbi12d |
⊢ ( 𝑥 = ( 𝐵 + 1 ) → ( ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ↔ ( 𝐵 < ( 𝐵 + 1 ) → ∃ 𝑦 ∈ 𝐴 𝑦 < ( 𝐵 + 1 ) ) ) ) |
89 |
88
|
rspcva |
⊢ ( ( ( 𝐵 + 1 ) ∈ ℝ ∧ ∀ 𝑥 ∈ ℝ ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) → ( 𝐵 < ( 𝐵 + 1 ) → ∃ 𝑦 ∈ 𝐴 𝑦 < ( 𝐵 + 1 ) ) ) |
90 |
83 84 89
|
sylc |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → ∃ 𝑦 ∈ 𝐴 𝑦 < ( 𝐵 + 1 ) ) |
91 |
|
nfv |
⊢ Ⅎ 𝑦 𝐵 < 𝑥 |
92 |
2 41 91
|
nf3an |
⊢ Ⅎ 𝑦 ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) |
93 |
|
nfv |
⊢ Ⅎ 𝑦 ¬ 𝐵 = -∞ |
94 |
92 93
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) |
95 |
43
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ 𝑦 ∈ 𝐴 ) → 𝑦 ∈ ℝ* ) |
96 |
95
|
ad4ant13 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < ( 𝐵 + 1 ) ) → 𝑦 ∈ ℝ* ) |
97 |
80
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐵 + 1 ) ∈ ℝ ) |
98 |
97
|
rexrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝐵 + 1 ) ∈ ℝ* ) |
99 |
98
|
adantr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < ( 𝐵 + 1 ) ) → ( 𝐵 + 1 ) ∈ ℝ* ) |
100 |
50
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) → 𝑥 ∈ ℝ* ) |
101 |
100
|
ad3antrrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < ( 𝐵 + 1 ) ) → 𝑥 ∈ ℝ* ) |
102 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < ( 𝐵 + 1 ) ) → 𝑦 < ( 𝐵 + 1 ) ) |
103 |
80
|
ltpnfd |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → ( 𝐵 + 1 ) < +∞ ) |
104 |
56
|
adantr |
⊢ ( ( 𝑥 = +∞ ∧ ¬ 𝐵 = -∞ ) → +∞ = 𝑥 ) |
105 |
104
|
3ad2antl2 |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → +∞ = 𝑥 ) |
106 |
103 105
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → ( 𝐵 + 1 ) < 𝑥 ) |
107 |
106
|
ad2antrr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < ( 𝐵 + 1 ) ) → ( 𝐵 + 1 ) < 𝑥 ) |
108 |
96 99 101 102 107
|
xrlttrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) ∧ 𝑦 ∈ 𝐴 ) ∧ 𝑦 < ( 𝐵 + 1 ) ) → 𝑦 < 𝑥 ) |
109 |
108
|
ex |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑦 < ( 𝐵 + 1 ) → 𝑦 < 𝑥 ) ) |
110 |
109
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → ( 𝑦 ∈ 𝐴 → ( 𝑦 < ( 𝐵 + 1 ) → 𝑦 < 𝑥 ) ) ) |
111 |
94 110
|
reximdai |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → ( ∃ 𝑦 ∈ 𝐴 𝑦 < ( 𝐵 + 1 ) → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) |
112 |
90 111
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) ∧ ¬ 𝐵 = -∞ ) → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) |
113 |
66 112
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑥 = +∞ ∧ 𝐵 < 𝑥 ) → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) |
114 |
9 23 24 113
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ ¬ 𝑥 ∈ ℝ ) ∧ 𝐵 < 𝑥 ) → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) |
115 |
114
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) ∧ ¬ 𝑥 ∈ ℝ ) → ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) |
116 |
8 115
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ* ) → ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) |
117 |
116
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ* → ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) ) |
118 |
1 117
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ* ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) |
119 |
|
xrltso |
⊢ < Or ℝ* |
120 |
119
|
a1i |
⊢ ( ⊤ → < Or ℝ* ) |
121 |
120
|
eqinf |
⊢ ( ⊤ → ( ( 𝐵 ∈ ℝ* ∧ ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵 ∧ ∀ 𝑥 ∈ ℝ* ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) → inf ( 𝐴 , ℝ* , < ) = 𝐵 ) ) |
122 |
121
|
mptru |
⊢ ( ( 𝐵 ∈ ℝ* ∧ ∀ 𝑥 ∈ 𝐴 ¬ 𝑥 < 𝐵 ∧ ∀ 𝑥 ∈ ℝ* ( 𝐵 < 𝑥 → ∃ 𝑦 ∈ 𝐴 𝑦 < 𝑥 ) ) → inf ( 𝐴 , ℝ* , < ) = 𝐵 ) |
123 |
4 5 118 122
|
syl3anc |
⊢ ( 𝜑 → inf ( 𝐴 , ℝ* , < ) = 𝐵 ) |