Proof of Theorem xrsupsslem
Step | Hyp | Ref
| Expression |
1 | | raleq 3341 |
. . . . . 6
⊢ (𝐴 = ∅ → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦)) |
2 | | rexeq 3342 |
. . . . . . . 8
⊢ (𝐴 = ∅ → (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ∃𝑧 ∈ ∅ 𝑦 < 𝑧)) |
3 | 2 | imbi2d 341 |
. . . . . . 7
⊢ (𝐴 = ∅ → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))) |
4 | 3 | ralbidv 3123 |
. . . . . 6
⊢ (𝐴 = ∅ → (∀𝑦 ∈ ℝ*
(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))) |
5 | 1, 4 | anbi12d 631 |
. . . . 5
⊢ (𝐴 = ∅ →
((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))) |
6 | 5 | rexbidv 3228 |
. . . 4
⊢ (𝐴 = ∅ → (∃𝑥 ∈ ℝ*
(∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) ↔ ∃𝑥 ∈ ℝ* (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)))) |
7 | | sup3 11924 |
. . . . . . . 8
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
8 | | rexr 11014 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℝ*) |
9 | 8 | anim1i 615 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧
(∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑥 ∈ ℝ* ∧
(∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
10 | 9 | reximi2 3174 |
. . . . . . . 8
⊢
(∃𝑥 ∈
ℝ (∀𝑦 ∈
𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
11 | 7, 10 | syl 17 |
. . . . . . 7
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
12 | | elxr 12843 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℝ*
↔ (𝑦 ∈ ℝ
∨ 𝑦 = +∞ ∨
𝑦 =
-∞)) |
13 | | simpr 485 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ (𝑦 ∈ ℝ
→ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑦 ∈ ℝ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
14 | | pnfnlt 12855 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ℝ*
→ ¬ +∞ < 𝑥) |
15 | 14 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 = +∞) →
¬ +∞ < 𝑥) |
16 | | breq1 5082 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = +∞ → (𝑦 < 𝑥 ↔ +∞ < 𝑥)) |
17 | 16 | notbid 318 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = +∞ → (¬ 𝑦 < 𝑥 ↔ ¬ +∞ < 𝑥)) |
18 | 17 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 = +∞) →
(¬ 𝑦 < 𝑥 ↔ ¬ +∞ <
𝑥)) |
19 | 15, 18 | mpbird 256 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 = +∞) →
¬ 𝑦 < 𝑥) |
20 | 19 | pm2.21d 121 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ*
∧ 𝑦 = +∞) →
(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) |
21 | 20 | ex 413 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℝ*
→ (𝑦 = +∞ →
(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
22 | 21 | ad2antlr 724 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ (𝑦 ∈ ℝ
→ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑦 = +∞ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
23 | | ssel 3919 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → 𝑧 ∈ ℝ)) |
24 | | mnflt 12850 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧 ∈ ℝ → -∞
< 𝑧) |
25 | 23, 24 | syl6 35 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → -∞ < 𝑧)) |
26 | 25 | ancld 551 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → (𝑧 ∈ 𝐴 ∧ -∞ < 𝑧))) |
27 | 26 | eximdv 1924 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ⊆ ℝ →
(∃𝑧 𝑧 ∈ 𝐴 → ∃𝑧(𝑧 ∈ 𝐴 ∧ -∞ < 𝑧))) |
28 | | n0 4286 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ≠ ∅ ↔
∃𝑧 𝑧 ∈ 𝐴) |
29 | | df-rex 3072 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∃𝑧 ∈
𝐴 -∞ < 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ -∞ < 𝑧)) |
30 | 27, 28, 29 | 3imtr4g 296 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ →
∃𝑧 ∈ 𝐴 -∞ < 𝑧)) |
31 | 30 | imp 407 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) →
∃𝑧 ∈ 𝐴 -∞ < 𝑧) |
32 | 31 | a1d 25 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (-∞
< 𝑥 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧)) |
33 | 32 | ad2antrr 723 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ 𝑦 = -∞) →
(-∞ < 𝑥 →
∃𝑧 ∈ 𝐴 -∞ < 𝑧)) |
34 | | breq1 5082 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = -∞ → (𝑦 < 𝑥 ↔ -∞ < 𝑥)) |
35 | | breq1 5082 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = -∞ → (𝑦 < 𝑧 ↔ -∞ < 𝑧)) |
36 | 35 | rexbidv 3228 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = -∞ → (∃𝑧 ∈ 𝐴 𝑦 < 𝑧 ↔ ∃𝑧 ∈ 𝐴 -∞ < 𝑧)) |
37 | 34, 36 | imbi12d 345 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = -∞ → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (-∞ < 𝑥 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧))) |
38 | 37 | adantl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ 𝑦 = -∞) →
((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (-∞ < 𝑥 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧))) |
39 | 33, 38 | mpbird 256 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ 𝑦 = -∞) →
(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) |
40 | 39 | ex 413 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
→ (𝑦 = -∞ →
(𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
41 | 40 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ (𝑦 ∈ ℝ
→ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑦 = -∞ → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
42 | 13, 22, 41 | 3jaod 1427 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ (𝑦 ∈ ℝ
→ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
43 | 12, 42 | syl5bi 241 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
∧ (𝑦 ∈ ℝ
→ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → (𝑦 ∈ ℝ* → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
44 | 43 | ex 413 |
. . . . . . . . . . 11
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
→ ((𝑦 ∈ ℝ
→ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → (𝑦 ∈ ℝ* → (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
45 | 44 | ralimdv2 3104 |
. . . . . . . . . 10
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
→ (∀𝑦 ∈
ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
46 | 45 | anim2d 612 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ ℝ*)
→ ((∀𝑦 ∈
𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
47 | 46 | reximdva 3205 |
. . . . . . . 8
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) →
(∃𝑥 ∈
ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
48 | 47 | 3adant3 1131 |
. . . . . . 7
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → (∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
49 | 11, 48 | mpd 15 |
. . . . . 6
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
50 | 49 | 3expa 1117 |
. . . . 5
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧
∃𝑥 ∈ ℝ
∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
51 | | ralnex 3166 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
ℝ ¬ ∀𝑦
∈ 𝐴 𝑦 ≤ 𝑥 ↔ ¬ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
52 | | rexnal 3168 |
. . . . . . . . . . . 12
⊢
(∃𝑦 ∈
𝐴 ¬ 𝑦 ≤ 𝑥 ↔ ¬ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
53 | | ssel2 3921 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ) |
54 | | letric 11067 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 ≤ 𝑥 ∨ 𝑥 ≤ 𝑦)) |
55 | 54 | ord 861 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (¬
𝑦 ≤ 𝑥 → 𝑥 ≤ 𝑦)) |
56 | 53, 55 | sylan 580 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ ℝ) → (¬ 𝑦 ≤ 𝑥 → 𝑥 ≤ 𝑦)) |
57 | 56 | an32s 649 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) ∧ 𝑦 ∈ 𝐴) → (¬ 𝑦 ≤ 𝑥 → 𝑥 ≤ 𝑦)) |
58 | 57 | reximdva 3205 |
. . . . . . . . . . . 12
⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) →
(∃𝑦 ∈ 𝐴 ¬ 𝑦 ≤ 𝑥 → ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) |
59 | 52, 58 | syl5bir 242 |
. . . . . . . . . . 11
⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (¬
∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) |
60 | 59 | ralimdva 3105 |
. . . . . . . . . 10
⊢ (𝐴 ⊆ ℝ →
(∀𝑥 ∈ ℝ
¬ ∀𝑦 ∈
𝐴 𝑦 ≤ 𝑥 → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦)) |
61 | 60 | imp 407 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
¬ ∀𝑦 ∈
𝐴 𝑦 ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
62 | 51, 61 | sylan2br 595 |
. . . . . . . 8
⊢ ((𝐴 ⊆ ℝ ∧ ¬
∃𝑥 ∈ ℝ
∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) |
63 | | breq2 5083 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝑧)) |
64 | 63 | cbvrexvw 3382 |
. . . . . . . . 9
⊢
(∃𝑦 ∈
𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) |
65 | 64 | ralbii 3093 |
. . . . . . . 8
⊢
(∀𝑥 ∈
ℝ ∃𝑦 ∈
𝐴 𝑥 ≤ 𝑦 ↔ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) |
66 | 62, 65 | sylib 217 |
. . . . . . 7
⊢ ((𝐴 ⊆ ℝ ∧ ¬
∃𝑥 ∈ ℝ
∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) |
67 | | pnfxr 11022 |
. . . . . . . 8
⊢ +∞
∈ ℝ* |
68 | | ssel 3919 |
. . . . . . . . . . . 12
⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ)) |
69 | | rexr 11014 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℝ → 𝑦 ∈
ℝ*) |
70 | | pnfnlt 12855 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℝ*
→ ¬ +∞ < 𝑦) |
71 | 69, 70 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℝ → ¬
+∞ < 𝑦) |
72 | 68, 71 | syl6 35 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → ¬ +∞ < 𝑦)) |
73 | 72 | ralrimiv 3109 |
. . . . . . . . . 10
⊢ (𝐴 ⊆ ℝ →
∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦) |
74 | 73 | adantr 481 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦) |
75 | | peano2re 11140 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈ ℝ → (𝑦 + 1) ∈
ℝ) |
76 | | breq1 5082 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = (𝑦 + 1) → (𝑥 ≤ 𝑧 ↔ (𝑦 + 1) ≤ 𝑧)) |
77 | 76 | rexbidv 3228 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = (𝑦 + 1) → (∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ↔ ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧)) |
78 | 77 | rspcva 3559 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑦 + 1) ∈ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧) |
79 | 78 | adantrr 714 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑦 + 1) ∈ ℝ ∧
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ)) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧) |
80 | 79 | ancoms 459 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) ∧ (𝑦 + 1) ∈ ℝ) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧) |
81 | 75, 80 | sylan2 593 |
. . . . . . . . . . . . . . . . . 18
⊢
(((∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧) |
82 | | ssel2 3921 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ) |
83 | | ltp1 11807 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ℝ → 𝑦 < (𝑦 + 1)) |
84 | 83 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑦 < (𝑦 + 1)) |
85 | 75 | ancli 549 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑦 ∈ ℝ → (𝑦 ∈ ℝ ∧ (𝑦 + 1) ∈
ℝ)) |
86 | | ltletr 11059 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑦 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦 < (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑧) → 𝑦 < 𝑧)) |
87 | 86 | 3expa 1117 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑦 ∈ ℝ ∧ (𝑦 + 1) ∈ ℝ) ∧
𝑧 ∈ ℝ) →
((𝑦 < (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑧) → 𝑦 < 𝑧)) |
88 | 85, 87 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦 < (𝑦 + 1) ∧ (𝑦 + 1) ≤ 𝑧) → 𝑦 < 𝑧)) |
89 | 84, 88 | mpand 692 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧)) |
90 | 89 | ancoms 459 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧)) |
91 | 82, 90 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) ∧ 𝑦 ∈ ℝ) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧)) |
92 | 91 | an32s 649 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑧 ∈ 𝐴) → ((𝑦 + 1) ≤ 𝑧 → 𝑦 < 𝑧)) |
93 | 92 | reximdva 3205 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ⊆ ℝ ∧ 𝑦 ∈ ℝ) →
(∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) |
94 | 93 | adantll 711 |
. . . . . . . . . . . . . . . . . 18
⊢
(((∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → (∃𝑧 ∈ 𝐴 (𝑦 + 1) ≤ 𝑧 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) |
95 | 81, 94 | mpd 15 |
. . . . . . . . . . . . . . . . 17
⊢
(((∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) ∧ 𝑦 ∈ ℝ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) |
96 | 95 | exp31 420 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 ∈ ℝ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
97 | 96 | a1dd 50 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → (𝑦 ∈ ℝ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
98 | 97 | com4r 94 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ ℝ →
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
99 | | xrltnr 12846 |
. . . . . . . . . . . . . . . . . 18
⊢ (+∞
∈ ℝ* → ¬ +∞ <
+∞) |
100 | 67, 99 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ ¬
+∞ < +∞ |
101 | | breq1 5082 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = +∞ → (𝑦 < +∞ ↔ +∞
< +∞)) |
102 | 100, 101 | mtbiri 327 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = +∞ → ¬ 𝑦 < +∞) |
103 | 102 | pm2.21d 121 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = +∞ → (𝑦 < +∞ →
∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) |
104 | 103 | 2a1d 26 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = +∞ →
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
105 | | 0re 10970 |
. . . . . . . . . . . . . . . . . . 19
⊢ 0 ∈
ℝ |
106 | | breq1 5082 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 0 → (𝑥 ≤ 𝑧 ↔ 0 ≤ 𝑧)) |
107 | 106 | rexbidv 3228 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 0 → (∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ↔ ∃𝑧 ∈ 𝐴 0 ≤ 𝑧)) |
108 | 107 | rspcva 3559 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((0
∈ ℝ ∧ ∀𝑥 ∈ ℝ ∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∃𝑧 ∈ 𝐴 0 ≤ 𝑧) |
109 | 105, 108 | mpan 687 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑥 ≤ 𝑧 → ∃𝑧 ∈ 𝐴 0 ≤ 𝑧) |
110 | 82, 24 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → -∞ < 𝑧) |
111 | 110 | a1d 25 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ⊆ ℝ ∧ 𝑧 ∈ 𝐴) → (0 ≤ 𝑧 → -∞ < 𝑧)) |
112 | 111 | reximdva 3205 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ⊆ ℝ →
(∃𝑧 ∈ 𝐴 0 ≤ 𝑧 → ∃𝑧 ∈ 𝐴 -∞ < 𝑧)) |
113 | 109, 112 | mpan9 507 |
. . . . . . . . . . . . . . . . 17
⊢
((∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) → ∃𝑧 ∈ 𝐴 -∞ < 𝑧) |
114 | 113, 36 | syl5ibr 245 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = -∞ →
((∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) |
115 | 114 | a1dd 50 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = -∞ →
((∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 ∧ 𝐴 ⊆ ℝ) → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
116 | 115 | expd 416 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = -∞ →
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
117 | 98, 104, 116 | 3jaoi 1426 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈ ℝ ∨ 𝑦 = +∞ ∨ 𝑦 = -∞) →
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
118 | 12, 117 | sylbi 216 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ℝ*
→ (∀𝑥 ∈
ℝ ∃𝑧 ∈
𝐴 𝑥 ≤ 𝑧 → (𝐴 ⊆ ℝ → (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
119 | 118 | com13 88 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ ℝ →
(∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧 → (𝑦 ∈ ℝ* → (𝑦 < +∞ →
∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
120 | 119 | imp 407 |
. . . . . . . . . 10
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → (𝑦 ∈ ℝ* → (𝑦 < +∞ →
∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
121 | 120 | ralrimiv 3109 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∀𝑦 ∈ ℝ* (𝑦 < +∞ →
∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) |
122 | 74, 121 | jca 512 |
. . . . . . . 8
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < +∞ →
∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
123 | | breq1 5082 |
. . . . . . . . . . . 12
⊢ (𝑥 = +∞ → (𝑥 < 𝑦 ↔ +∞ < 𝑦)) |
124 | 123 | notbid 318 |
. . . . . . . . . . 11
⊢ (𝑥 = +∞ → (¬ 𝑥 < 𝑦 ↔ ¬ +∞ < 𝑦)) |
125 | 124 | ralbidv 3123 |
. . . . . . . . . 10
⊢ (𝑥 = +∞ →
(∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦)) |
126 | | breq2 5083 |
. . . . . . . . . . . 12
⊢ (𝑥 = +∞ → (𝑦 < 𝑥 ↔ 𝑦 < +∞)) |
127 | 126 | imbi1d 342 |
. . . . . . . . . . 11
⊢ (𝑥 = +∞ → ((𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ (𝑦 < +∞ → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
128 | 127 | ralbidv 3123 |
. . . . . . . . . 10
⊢ (𝑥 = +∞ →
(∀𝑦 ∈
ℝ* (𝑦 <
𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ* (𝑦 < +∞ →
∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
129 | 125, 128 | anbi12d 631 |
. . . . . . . . 9
⊢ (𝑥 = +∞ →
((∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < +∞ →
∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))) |
130 | 129 | rspcev 3561 |
. . . . . . . 8
⊢
((+∞ ∈ ℝ* ∧ (∀𝑦 ∈ 𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < +∞ →
∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
131 | 67, 122, 130 | sylancr 587 |
. . . . . . 7
⊢ ((𝐴 ⊆ ℝ ∧
∀𝑥 ∈ ℝ
∃𝑧 ∈ 𝐴 𝑥 ≤ 𝑧) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
132 | 66, 131 | syldan 591 |
. . . . . 6
⊢ ((𝐴 ⊆ ℝ ∧ ¬
∃𝑥 ∈ ℝ
∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
133 | 132 | adantlr 712 |
. . . . 5
⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ¬
∃𝑥 ∈ ℝ
∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
134 | 50, 133 | pm2.61dan 810 |
. . . 4
⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) →
∃𝑥 ∈
ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
135 | | mnfxr 11025 |
. . . . . 6
⊢ -∞
∈ ℝ* |
136 | | ral0 4449 |
. . . . . . 7
⊢
∀𝑦 ∈
∅ ¬ -∞ < 𝑦 |
137 | | nltmnf 12856 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℝ*
→ ¬ 𝑦 <
-∞) |
138 | 137 | pm2.21d 121 |
. . . . . . . 8
⊢ (𝑦 ∈ ℝ*
→ (𝑦 < -∞
→ ∃𝑧 ∈
∅ 𝑦 < 𝑧)) |
139 | 138 | rgen 3076 |
. . . . . . 7
⊢
∀𝑦 ∈
ℝ* (𝑦 <
-∞ → ∃𝑧
∈ ∅ 𝑦 < 𝑧) |
140 | 136, 139 | pm3.2i 471 |
. . . . . 6
⊢
(∀𝑦 ∈
∅ ¬ -∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < -∞ →
∃𝑧 ∈ ∅
𝑦 < 𝑧)) |
141 | | breq1 5082 |
. . . . . . . . . 10
⊢ (𝑥 = -∞ → (𝑥 < 𝑦 ↔ -∞ < 𝑦)) |
142 | 141 | notbid 318 |
. . . . . . . . 9
⊢ (𝑥 = -∞ → (¬ 𝑥 < 𝑦 ↔ ¬ -∞ < 𝑦)) |
143 | 142 | ralbidv 3123 |
. . . . . . . 8
⊢ (𝑥 = -∞ →
(∀𝑦 ∈ ∅
¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ ∅ ¬ -∞
< 𝑦)) |
144 | | breq2 5083 |
. . . . . . . . . 10
⊢ (𝑥 = -∞ → (𝑦 < 𝑥 ↔ 𝑦 < -∞)) |
145 | 144 | imbi1d 342 |
. . . . . . . . 9
⊢ (𝑥 = -∞ → ((𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧) ↔ (𝑦 < -∞ → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))) |
146 | 145 | ralbidv 3123 |
. . . . . . . 8
⊢ (𝑥 = -∞ →
(∀𝑦 ∈
ℝ* (𝑦 <
𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ* (𝑦 < -∞ →
∃𝑧 ∈ ∅
𝑦 < 𝑧))) |
147 | 143, 146 | anbi12d 631 |
. . . . . . 7
⊢ (𝑥 = -∞ →
((∀𝑦 ∈ ∅
¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ*
(𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)) ↔ (∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ*
(𝑦 < -∞ →
∃𝑧 ∈ ∅
𝑦 < 𝑧)))) |
148 | 147 | rspcev 3561 |
. . . . . 6
⊢
((-∞ ∈ ℝ* ∧ (∀𝑦 ∈ ∅ ¬ -∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ*
(𝑦 < -∞ →
∃𝑧 ∈ ∅
𝑦 < 𝑧))) → ∃𝑥 ∈ ℝ* (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))) |
149 | 135, 140,
148 | mp2an 689 |
. . . . 5
⊢
∃𝑥 ∈
ℝ* (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧)) |
150 | 149 | a1i 11 |
. . . 4
⊢ (𝐴 ⊆ ℝ →
∃𝑥 ∈
ℝ* (∀𝑦 ∈ ∅ ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ ∅ 𝑦 < 𝑧))) |
151 | 6, 134, 150 | pm2.61ne 3032 |
. . 3
⊢ (𝐴 ⊆ ℝ →
∃𝑥 ∈
ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
152 | 151 | adantl 482 |
. 2
⊢ ((𝐴 ⊆ ℝ*
∧ 𝐴 ⊆ ℝ)
→ ∃𝑥 ∈
ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
153 | | ssel 3919 |
. . . . . 6
⊢ (𝐴 ⊆ ℝ*
→ (𝑦 ∈ 𝐴 → 𝑦 ∈
ℝ*)) |
154 | 153, 70 | syl6 35 |
. . . . 5
⊢ (𝐴 ⊆ ℝ*
→ (𝑦 ∈ 𝐴 → ¬ +∞ <
𝑦)) |
155 | 154 | ralrimiv 3109 |
. . . 4
⊢ (𝐴 ⊆ ℝ*
→ ∀𝑦 ∈
𝐴 ¬ +∞ < 𝑦) |
156 | | breq2 5083 |
. . . . . . 7
⊢ (𝑧 = +∞ → (𝑦 < 𝑧 ↔ 𝑦 < +∞)) |
157 | 156 | rspcev 3561 |
. . . . . 6
⊢
((+∞ ∈ 𝐴
∧ 𝑦 < +∞)
→ ∃𝑧 ∈
𝐴 𝑦 < 𝑧) |
158 | 157 | ex 413 |
. . . . 5
⊢ (+∞
∈ 𝐴 → (𝑦 < +∞ →
∃𝑧 ∈ 𝐴 𝑦 < 𝑧)) |
159 | 158 | ralrimivw 3111 |
. . . 4
⊢ (+∞
∈ 𝐴 →
∀𝑦 ∈
ℝ* (𝑦 <
+∞ → ∃𝑧
∈ 𝐴 𝑦 < 𝑧)) |
160 | 155, 159 | anim12i 613 |
. . 3
⊢ ((𝐴 ⊆ ℝ*
∧ +∞ ∈ 𝐴)
→ (∀𝑦 ∈
𝐴 ¬ +∞ < 𝑦 ∧ ∀𝑦 ∈ ℝ*
(𝑦 < +∞ →
∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
161 | 67, 160, 130 | sylancr 587 |
. 2
⊢ ((𝐴 ⊆ ℝ*
∧ +∞ ∈ 𝐴)
→ ∃𝑥 ∈
ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |
162 | 152, 161 | jaodan 955 |
1
⊢ ((𝐴 ⊆ ℝ*
∧ (𝐴 ⊆ ℝ
∨ +∞ ∈ 𝐴))
→ ∃𝑥 ∈
ℝ* (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ* (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧))) |