Step | Hyp | Ref
| Expression |
1 | | breq1 4586 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑀 → (ℎ ≤ 𝑡 ↔ 𝑀 ≤ 𝑡)) |
2 | 1 | ralbidv 2969 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑀 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡)) |
3 | 2 | imbi2d 329 |
. . . . . . . . . 10
⊢ (ℎ = 𝑀 → (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡) ↔ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡))) |
4 | | breq1 4586 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑚 → (ℎ ≤ 𝑡 ↔ 𝑚 ≤ 𝑡)) |
5 | 4 | ralbidv 2969 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑚 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡)) |
6 | 5 | imbi2d 329 |
. . . . . . . . . 10
⊢ (ℎ = 𝑚 → (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡) ↔ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡))) |
7 | | breq1 4586 |
. . . . . . . . . . . 12
⊢ (ℎ = (𝑚 + 1) → (ℎ ≤ 𝑡 ↔ (𝑚 + 1) ≤ 𝑡)) |
8 | 7 | ralbidv 2969 |
. . . . . . . . . . 11
⊢ (ℎ = (𝑚 + 1) → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)) |
9 | 8 | imbi2d 329 |
. . . . . . . . . 10
⊢ (ℎ = (𝑚 + 1) → (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡) ↔ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
10 | | breq1 4586 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑛 → (ℎ ≤ 𝑡 ↔ 𝑛 ≤ 𝑡)) |
11 | 10 | ralbidv 2969 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑛 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡)) |
12 | 11 | imbi2d 329 |
. . . . . . . . . 10
⊢ (ℎ = 𝑛 → (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡) ↔ ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡))) |
13 | | ssel 3562 |
. . . . . . . . . . . . . 14
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (𝑡 ∈ 𝑆 → 𝑡 ∈ (ℤ≥‘𝑀))) |
14 | | eluzle 11576 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈
(ℤ≥‘𝑀) → 𝑀 ≤ 𝑡) |
15 | 13, 14 | syl6 34 |
. . . . . . . . . . . . 13
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (𝑡 ∈ 𝑆 → 𝑀 ≤ 𝑡)) |
16 | 15 | ralrimiv 2948 |
. . . . . . . . . . . 12
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡) |
17 | 16 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡) |
18 | 17 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℤ → ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡)) |
19 | | uzssz 11583 |
. . . . . . . . . . . . 13
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
20 | | sstr 3576 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ (ℤ≥‘𝑀) ⊆ ℤ) → 𝑆 ⊆
ℤ) |
21 | 19, 20 | mpan2 703 |
. . . . . . . . . . . 12
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → 𝑆 ⊆ ℤ) |
22 | | eluzelz 11573 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈
(ℤ≥‘𝑀) → 𝑚 ∈ ℤ) |
23 | | breq1 4586 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑚 → (𝑗 ≤ 𝑡 ↔ 𝑚 ≤ 𝑡)) |
24 | 23 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑚 → (∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡)) |
25 | 24 | rspcev 3282 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑚 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) |
26 | 25 | expcom 450 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑡 ∈
𝑆 𝑚 ≤ 𝑡 → (𝑚 ∈ 𝑆 → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)) |
27 | 26 | con3rr3 150 |
. . . . . . . . . . . . . . . 16
⊢ (¬
∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ¬ 𝑚 ∈ 𝑆)) |
28 | | ssel2 3563 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆) → 𝑡 ∈ ℤ) |
29 | | zre 11258 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑚 ∈ ℤ → 𝑚 ∈
ℝ) |
30 | | zre 11258 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 ∈ ℤ → 𝑡 ∈
ℝ) |
31 | | letri3 10002 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑚 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚))) |
32 | 29, 30, 31 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚))) |
33 | | zleltp1 11305 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ 𝑡 < (𝑚 + 1))) |
34 | | peano2re 10088 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑚 ∈ ℝ → (𝑚 + 1) ∈
ℝ) |
35 | 29, 34 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ ℤ → (𝑚 + 1) ∈
ℝ) |
36 | | ltnle 9996 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑡 ∈ ℝ ∧ (𝑚 + 1) ∈ ℝ) →
(𝑡 < (𝑚 + 1) ↔ ¬ (𝑚 + 1) ≤ 𝑡)) |
37 | 30, 35, 36 | syl2an 493 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 < (𝑚 + 1) ↔ ¬ (𝑚 + 1) ≤ 𝑡)) |
38 | 33, 37 | bitrd 267 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ ¬ (𝑚 + 1) ≤ 𝑡)) |
39 | 38 | ancoms 468 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ ¬ (𝑚 + 1) ≤ 𝑡)) |
40 | 39 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → ((𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚) ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡))) |
41 | 32, 40 | bitrd 267 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡))) |
42 | 28, 41 | sylan2 490 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡))) |
43 | | eleq1a 2683 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 ∈ 𝑆 → (𝑚 = 𝑡 → 𝑚 ∈ 𝑆)) |
44 | 43 | ad2antll 761 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 = 𝑡 → 𝑚 ∈ 𝑆)) |
45 | 42, 44 | sylbird 249 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → ((𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡) → 𝑚 ∈ 𝑆)) |
46 | 45 | expd 451 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 ≤ 𝑡 → (¬ (𝑚 + 1) ≤ 𝑡 → 𝑚 ∈ 𝑆))) |
47 | | con1 142 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((¬
(𝑚 + 1) ≤ 𝑡 → 𝑚 ∈ 𝑆) → (¬ 𝑚 ∈ 𝑆 → (𝑚 + 1) ≤ 𝑡)) |
48 | 46, 47 | syl6 34 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 ≤ 𝑡 → (¬ 𝑚 ∈ 𝑆 → (𝑚 + 1) ≤ 𝑡))) |
49 | 48 | com23 84 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (¬ 𝑚 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡))) |
50 | 49 | exp32 629 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 ∈ ℤ → (𝑆 ⊆ ℤ → (𝑡 ∈ 𝑆 → (¬ 𝑚 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡))))) |
51 | 50 | com34 89 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈ ℤ → (𝑆 ⊆ ℤ → (¬
𝑚 ∈ 𝑆 → (𝑡 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡))))) |
52 | 51 | imp41 617 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬
𝑚 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡)) |
53 | 52 | ralimdva 2945 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬
𝑚 ∈ 𝑆) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)) |
54 | 53 | ex 449 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) → (¬
𝑚 ∈ 𝑆 → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
55 | 27, 54 | sylan9r 688 |
. . . . . . . . . . . . . . 15
⊢ (((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬
∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
56 | 55 | pm2.43d 51 |
. . . . . . . . . . . . . 14
⊢ (((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬
∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)) |
57 | 56 | expl 646 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℤ → ((𝑆 ⊆ ℤ ∧ ¬
∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
58 | 22, 57 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈
(ℤ≥‘𝑀) → ((𝑆 ⊆ ℤ ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
59 | 21, 58 | sylani 684 |
. . . . . . . . . . 11
⊢ (𝑚 ∈
(ℤ≥‘𝑀) → ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
60 | 59 | a2d 29 |
. . . . . . . . . 10
⊢ (𝑚 ∈
(ℤ≥‘𝑀) → (((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡) → ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
61 | 3, 6, 9, 12, 18, 60 | uzind4 11622 |
. . . . . . . . 9
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡)) |
62 | | breq1 4586 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑛 → (𝑗 ≤ 𝑡 ↔ 𝑛 ≤ 𝑡)) |
63 | 62 | ralbidv 2969 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑛 → (∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡)) |
64 | 63 | rspcev 3282 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) |
65 | 64 | expcom 450 |
. . . . . . . . . . 11
⊢
(∀𝑡 ∈
𝑆 𝑛 ≤ 𝑡 → (𝑛 ∈ 𝑆 → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)) |
66 | 65 | con3rr3 150 |
. . . . . . . . . 10
⊢ (¬
∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → (∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡 → ¬ 𝑛 ∈ 𝑆)) |
67 | 66 | adantl 481 |
. . . . . . . . 9
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡 → ¬ 𝑛 ∈ 𝑆)) |
68 | 61, 67 | sylcom 30 |
. . . . . . . 8
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ¬ 𝑛 ∈ 𝑆)) |
69 | | ssel 3562 |
. . . . . . . . . 10
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (𝑛 ∈ 𝑆 → 𝑛 ∈ (ℤ≥‘𝑀))) |
70 | 69 | con3rr3 150 |
. . . . . . . . 9
⊢ (¬
𝑛 ∈
(ℤ≥‘𝑀) → (𝑆 ⊆ (ℤ≥‘𝑀) → ¬ 𝑛 ∈ 𝑆)) |
71 | 70 | adantrd 483 |
. . . . . . . 8
⊢ (¬
𝑛 ∈
(ℤ≥‘𝑀) → ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ¬ 𝑛 ∈ 𝑆)) |
72 | 68, 71 | pm2.61i 175 |
. . . . . . 7
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ¬ 𝑛 ∈ 𝑆) |
73 | 72 | ex 449 |
. . . . . 6
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → ¬ 𝑛 ∈ 𝑆)) |
74 | 73 | alrimdv 1844 |
. . . . 5
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → ∀𝑛 ¬ 𝑛 ∈ 𝑆)) |
75 | | eq0 3888 |
. . . . 5
⊢ (𝑆 = ∅ ↔ ∀𝑛 ¬ 𝑛 ∈ 𝑆) |
76 | 74, 75 | syl6ibr 241 |
. . . 4
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → 𝑆 = ∅)) |
77 | 76 | necon1ad 2799 |
. . 3
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (𝑆 ≠ ∅ → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)) |
78 | 77 | imp 444 |
. 2
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) |
79 | | breq2 4587 |
. . . 4
⊢ (𝑡 = 𝑘 → (𝑗 ≤ 𝑡 ↔ 𝑗 ≤ 𝑘)) |
80 | 79 | cbvralv 3147 |
. . 3
⊢
(∀𝑡 ∈
𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) |
81 | 80 | rexbii 3023 |
. 2
⊢
(∃𝑗 ∈
𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) |
82 | 78, 81 | sylib 207 |
1
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) |