Step | Hyp | Ref
| Expression |
1 | | breq1 5056 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑀 → (ℎ ≤ 𝑡 ↔ 𝑀 ≤ 𝑡)) |
2 | 1 | ralbidv 3118 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑀 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡)) |
3 | 2 | imbi2d 344 |
. . . . . . . . . 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 5056 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑚 → (ℎ ≤ 𝑡 ↔ 𝑚 ≤ 𝑡)) |
5 | 4 | ralbidv 3118 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑚 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡)) |
6 | 5 | imbi2d 344 |
. . . . . . . . . 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 5056 |
. . . . . . . . . . . 12
⊢ (ℎ = (𝑚 + 1) → (ℎ ≤ 𝑡 ↔ (𝑚 + 1) ≤ 𝑡)) |
8 | 7 | ralbidv 3118 |
. . . . . . . . . . 11
⊢ (ℎ = (𝑚 + 1) → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)) |
9 | 8 | imbi2d 344 |
. . . . . . . . . 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 5056 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑛 → (ℎ ≤ 𝑡 ↔ 𝑛 ≤ 𝑡)) |
11 | 10 | ralbidv 3118 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑛 → (∀𝑡 ∈ 𝑆 ℎ ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡)) |
12 | 11 | imbi2d 344 |
. . . . . . . . . 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 3893 |
. . . . . . . . . . . . 13
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (𝑡 ∈ 𝑆 → 𝑡 ∈ (ℤ≥‘𝑀))) |
14 | | eluzle 12451 |
. . . . . . . . . . . . 13
⊢ (𝑡 ∈
(ℤ≥‘𝑀) → 𝑀 ≤ 𝑡) |
15 | 13, 14 | syl6 35 |
. . . . . . . . . . . 12
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (𝑡 ∈ 𝑆 → 𝑀 ≤ 𝑡)) |
16 | 15 | ralrimiv 3104 |
. . . . . . . . . . 11
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡) |
17 | 16 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑀 ≤ 𝑡) |
18 | | uzssz 12459 |
. . . . . . . . . . . . 13
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
19 | | sstr 3909 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ (ℤ≥‘𝑀) ⊆ ℤ) → 𝑆 ⊆
ℤ) |
20 | 18, 19 | mpan2 691 |
. . . . . . . . . . . 12
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → 𝑆 ⊆ ℤ) |
21 | | eluzelz 12448 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈
(ℤ≥‘𝑀) → 𝑚 ∈ ℤ) |
22 | | breq1 5056 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑚 → (𝑗 ≤ 𝑡 ↔ 𝑚 ≤ 𝑡)) |
23 | 22 | ralbidv 3118 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑚 → (∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡)) |
24 | 23 | rspcev 3537 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑚 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) |
25 | 24 | expcom 417 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑡 ∈
𝑆 𝑚 ≤ 𝑡 → (𝑚 ∈ 𝑆 → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)) |
26 | 25 | con3rr3 158 |
. . . . . . . . . . . . . . . 16
⊢ (¬
∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ¬ 𝑚 ∈ 𝑆)) |
27 | | ssel2 3895 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆) → 𝑡 ∈ ℤ) |
28 | | zre 12180 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑚 ∈ ℤ → 𝑚 ∈
ℝ) |
29 | | zre 12180 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑡 ∈ ℤ → 𝑡 ∈
ℝ) |
30 | | letri3 10918 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑚 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚))) |
31 | 28, 29, 30 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚))) |
32 | | zleltp1 12228 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ 𝑡 < (𝑚 + 1))) |
33 | | peano2re 11005 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑚 ∈ ℝ → (𝑚 + 1) ∈
ℝ) |
34 | 28, 33 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑚 ∈ ℤ → (𝑚 + 1) ∈
ℝ) |
35 | | ltnle 10912 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑡 ∈ ℝ ∧ (𝑚 + 1) ∈ ℝ) →
(𝑡 < (𝑚 + 1) ↔ ¬ (𝑚 + 1) ≤ 𝑡)) |
36 | 29, 34, 35 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 < (𝑚 + 1) ↔ ¬ (𝑚 + 1) ≤ 𝑡)) |
37 | 32, 36 | bitrd 282 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑡 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ ¬ (𝑚 + 1) ≤ 𝑡)) |
38 | 37 | ancoms 462 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑡 ≤ 𝑚 ↔ ¬ (𝑚 + 1) ≤ 𝑡)) |
39 | 38 | anbi2d 632 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → ((𝑚 ≤ 𝑡 ∧ 𝑡 ≤ 𝑚) ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡))) |
40 | 31, 39 | bitrd 282 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑚 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡))) |
41 | 27, 40 | sylan2 596 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 = 𝑡 ↔ (𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡))) |
42 | | eleq1a 2833 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑡 ∈ 𝑆 → (𝑚 = 𝑡 → 𝑚 ∈ 𝑆)) |
43 | 42 | ad2antll 729 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 = 𝑡 → 𝑚 ∈ 𝑆)) |
44 | 41, 43 | sylbird 263 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → ((𝑚 ≤ 𝑡 ∧ ¬ (𝑚 + 1) ≤ 𝑡) → 𝑚 ∈ 𝑆)) |
45 | 44 | expd 419 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 ≤ 𝑡 → (¬ (𝑚 + 1) ≤ 𝑡 → 𝑚 ∈ 𝑆))) |
46 | | con1 148 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((¬
(𝑚 + 1) ≤ 𝑡 → 𝑚 ∈ 𝑆) → (¬ 𝑚 ∈ 𝑆 → (𝑚 + 1) ≤ 𝑡)) |
47 | 45, 46 | syl6 35 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (𝑚 ≤ 𝑡 → (¬ 𝑚 ∈ 𝑆 → (𝑚 + 1) ≤ 𝑡))) |
48 | 47 | com23 86 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑚 ∈ ℤ ∧ (𝑆 ⊆ ℤ ∧ 𝑡 ∈ 𝑆)) → (¬ 𝑚 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡))) |
49 | 48 | exp32 424 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑚 ∈ ℤ → (𝑆 ⊆ ℤ → (𝑡 ∈ 𝑆 → (¬ 𝑚 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡))))) |
50 | 49 | com34 91 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈ ℤ → (𝑆 ⊆ ℤ → (¬
𝑚 ∈ 𝑆 → (𝑡 ∈ 𝑆 → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡))))) |
51 | 50 | imp41 429 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬
𝑚 ∈ 𝑆) ∧ 𝑡 ∈ 𝑆) → (𝑚 ≤ 𝑡 → (𝑚 + 1) ≤ 𝑡)) |
52 | 51 | ralimdva 3100 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬
𝑚 ∈ 𝑆) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)) |
53 | 52 | ex 416 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) → (¬
𝑚 ∈ 𝑆 → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
54 | 26, 53 | sylan9r 512 |
. . . . . . . . . . . . . . 15
⊢ (((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬
∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
55 | 54 | pm2.43d 53 |
. . . . . . . . . . . . . 14
⊢ (((𝑚 ∈ ℤ ∧ 𝑆 ⊆ ℤ) ∧ ¬
∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡)) |
56 | 55 | expl 461 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℤ → ((𝑆 ⊆ ℤ ∧ ¬
∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
57 | 21, 56 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈
(ℤ≥‘𝑀) → ((𝑆 ⊆ ℤ ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
58 | 20, 57 | sylani 607 |
. . . . . . . . . . 11
⊢ (𝑚 ∈
(ℤ≥‘𝑀) → ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑚 ≤ 𝑡 → ∀𝑡 ∈ 𝑆 (𝑚 + 1) ≤ 𝑡))) |
59 | 58 | 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))) |
60 | 3, 6, 9, 12, 17, 59 | uzind4i 12506 |
. . . . . . . . 9
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡)) |
61 | | breq1 5056 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑛 → (𝑗 ≤ 𝑡 ↔ 𝑛 ≤ 𝑡)) |
62 | 61 | ralbidv 3118 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑛 → (∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡)) |
63 | 62 | rspcev 3537 |
. . . . . . . . . . . 12
⊢ ((𝑛 ∈ 𝑆 ∧ ∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) |
64 | 63 | expcom 417 |
. . . . . . . . . . 11
⊢
(∀𝑡 ∈
𝑆 𝑛 ≤ 𝑡 → (𝑛 ∈ 𝑆 → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)) |
65 | 64 | con3rr3 158 |
. . . . . . . . . 10
⊢ (¬
∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → (∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡 → ¬ 𝑛 ∈ 𝑆)) |
66 | 65 | adantl 485 |
. . . . . . . . 9
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → (∀𝑡 ∈ 𝑆 𝑛 ≤ 𝑡 → ¬ 𝑛 ∈ 𝑆)) |
67 | 60, 66 | sylcom 30 |
. . . . . . . 8
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ¬ 𝑛 ∈ 𝑆)) |
68 | | ssel 3893 |
. . . . . . . . . 10
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (𝑛 ∈ 𝑆 → 𝑛 ∈ (ℤ≥‘𝑀))) |
69 | 68 | con3rr3 158 |
. . . . . . . . 9
⊢ (¬
𝑛 ∈
(ℤ≥‘𝑀) → (𝑆 ⊆ (ℤ≥‘𝑀) → ¬ 𝑛 ∈ 𝑆)) |
70 | 69 | adantrd 495 |
. . . . . . . 8
⊢ (¬
𝑛 ∈
(ℤ≥‘𝑀) → ((𝑆 ⊆ (ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ¬ 𝑛 ∈ 𝑆)) |
71 | 67, 70 | pm2.61i 185 |
. . . . . . 7
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) → ¬ 𝑛 ∈ 𝑆) |
72 | 71 | ex 416 |
. . . . . 6
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → ¬ 𝑛 ∈ 𝑆)) |
73 | 72 | alrimdv 1937 |
. . . . 5
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → ∀𝑛 ¬ 𝑛 ∈ 𝑆)) |
74 | | eq0 4258 |
. . . . 5
⊢ (𝑆 = ∅ ↔ ∀𝑛 ¬ 𝑛 ∈ 𝑆) |
75 | 73, 74 | syl6ibr 255 |
. . . 4
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (¬ ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 → 𝑆 = ∅)) |
76 | 75 | necon1ad 2957 |
. . 3
⊢ (𝑆 ⊆
(ℤ≥‘𝑀) → (𝑆 ≠ ∅ → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡)) |
77 | 76 | imp 410 |
. 2
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡) |
78 | | breq2 5057 |
. . . 4
⊢ (𝑡 = 𝑘 → (𝑗 ≤ 𝑡 ↔ 𝑗 ≤ 𝑘)) |
79 | 78 | cbvralvw 3358 |
. . 3
⊢
(∀𝑡 ∈
𝑆 𝑗 ≤ 𝑡 ↔ ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) |
80 | 79 | rexbii 3170 |
. 2
⊢
(∃𝑗 ∈
𝑆 ∀𝑡 ∈ 𝑆 𝑗 ≤ 𝑡 ↔ ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) |
81 | 77, 80 | sylib 221 |
1
⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ 𝑆 ≠ ∅) → ∃𝑗 ∈ 𝑆 ∀𝑘 ∈ 𝑆 𝑗 ≤ 𝑘) |