Step | Hyp | Ref
| Expression |
1 | | nfcv 2751 |
. . . . . . . . 9
⊢
Ⅎ𝑚𝐴 |
2 | | nfcsb1v 3515 |
. . . . . . . . 9
⊢
Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 |
3 | | csbeq1a 3508 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴) |
4 | 1, 2, 3 | cbviun 4493 |
. . . . . . . 8
⊢ ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴 |
5 | 4 | eleq2i 2680 |
. . . . . . 7
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ 𝑥 ∈ ∪
𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴) |
6 | | eliun 4460 |
. . . . . . 7
⊢ (𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) |
7 | 5, 6 | bitri 263 |
. . . . . 6
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) |
8 | 7 | biimpi 205 |
. . . . 5
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 → ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) |
9 | 8 | adantl 481 |
. . . 4
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐴) → ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) |
10 | | nfra1 2925 |
. . . . . 6
⊢
Ⅎ𝑚∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 |
11 | | nfv 1830 |
. . . . . 6
⊢
Ⅎ𝑚 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵 |
12 | | simp2 1055 |
. . . . . . . 8
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑚 ∈ 𝑍) |
13 | | rspa 2914 |
. . . . . . . . 9
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍) → ∪
𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵) |
14 | 13 | 3adant3 1074 |
. . . . . . . 8
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∪
𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵) |
15 | | simp3 1056 |
. . . . . . . 8
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) |
16 | | id 22 |
. . . . . . . . . . 11
⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪
𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵) |
17 | | fzssuz 12253 |
. . . . . . . . . . . . 13
⊢ (𝑁...𝑚) ⊆ (ℤ≥‘𝑁) |
18 | | iuneqfzuzlem.z |
. . . . . . . . . . . . . 14
⊢ 𝑍 =
(ℤ≥‘𝑁) |
19 | 18 | eqcomi 2619 |
. . . . . . . . . . . . 13
⊢
(ℤ≥‘𝑁) = 𝑍 |
20 | 17, 19 | sseqtri 3600 |
. . . . . . . . . . . 12
⊢ (𝑁...𝑚) ⊆ 𝑍 |
21 | | iunss1 4468 |
. . . . . . . . . . . 12
⊢ ((𝑁...𝑚) ⊆ 𝑍 → ∪
𝑛 ∈ (𝑁...𝑚)𝐵 ⊆ ∪
𝑛 ∈ 𝑍 𝐵) |
22 | 20, 21 | mp1i 13 |
. . . . . . . . . . 11
⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪
𝑛 ∈ (𝑁...𝑚)𝐵 ⊆ ∪
𝑛 ∈ 𝑍 𝐵) |
23 | 16, 22 | eqsstrd 3602 |
. . . . . . . . . 10
⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪
𝑛 ∈ (𝑁...𝑚)𝐴 ⊆ ∪
𝑛 ∈ 𝑍 𝐵) |
24 | 23 | 3ad2ant2 1076 |
. . . . . . . . 9
⊢ ((𝑚 ∈ 𝑍 ∧ ∪
𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∪
𝑛 ∈ (𝑁...𝑚)𝐴 ⊆ ∪
𝑛 ∈ 𝑍 𝐵) |
25 | 18 | eleq2i 2680 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ 𝑍 ↔ 𝑚 ∈ (ℤ≥‘𝑁)) |
26 | 25 | biimpi 205 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (ℤ≥‘𝑁)) |
27 | | eluzel2 11568 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈
(ℤ≥‘𝑁) → 𝑁 ∈ ℤ) |
28 | 26, 27 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ 𝑍 → 𝑁 ∈ ℤ) |
29 | | eluzelz 11573 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈
(ℤ≥‘𝑁) → 𝑚 ∈ ℤ) |
30 | 26, 29 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ) |
31 | 28, 30, 30 | 3jca 1235 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ 𝑍 → (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ)) |
32 | | eluzle 11576 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈
(ℤ≥‘𝑁) → 𝑁 ≤ 𝑚) |
33 | 26, 32 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ 𝑍 → 𝑁 ≤ 𝑚) |
34 | 30 | zred 11358 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℝ) |
35 | | leid 10012 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℝ → 𝑚 ≤ 𝑚) |
36 | 34, 35 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ 𝑍 → 𝑚 ≤ 𝑚) |
37 | 31, 33, 36 | jca32 556 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ 𝑍 → ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑁 ≤ 𝑚 ∧ 𝑚 ≤ 𝑚))) |
38 | | elfz2 12204 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ (𝑁...𝑚) ↔ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (𝑁 ≤ 𝑚 ∧ 𝑚 ≤ 𝑚))) |
39 | 37, 38 | sylibr 223 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (𝑁...𝑚)) |
40 | | nfcv 2751 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛𝑥 |
41 | 40, 2 | nfel 2763 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 |
42 | 3 | eleq2d 2673 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)) |
43 | 41, 42 | rspce 3277 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ (𝑁...𝑚) ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴) |
44 | 39, 43 | sylan 487 |
. . . . . . . . . . 11
⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴) |
45 | | eliun 4460 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 ↔ ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴) |
46 | 44, 45 | sylibr 223 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪
𝑛 ∈ (𝑁...𝑚)𝐴) |
47 | 46 | 3adant2 1073 |
. . . . . . . . 9
⊢ ((𝑚 ∈ 𝑍 ∧ ∪
𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪
𝑛 ∈ (𝑁...𝑚)𝐴) |
48 | 24, 47 | sseldd 3569 |
. . . . . . . 8
⊢ ((𝑚 ∈ 𝑍 ∧ ∪
𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
49 | 12, 14, 15, 48 | syl3anc 1318 |
. . . . . . 7
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
50 | 49 | 3exp 1256 |
. . . . . 6
⊢
(∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → (𝑚 ∈ 𝑍 → (𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵))) |
51 | 10, 11, 50 | rexlimd 3008 |
. . . . 5
⊢
(∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → (∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵)) |
52 | 51 | adantr 480 |
. . . 4
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐴) → (∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵)) |
53 | 9, 52 | mpd 15 |
. . 3
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐴) → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
54 | 53 | ralrimiva 2949 |
. 2
⊢
(∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
55 | | dfss3 3558 |
. 2
⊢ (∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪
𝑛 ∈ 𝑍 𝐵 ↔ ∀𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
56 | 54, 55 | sylibr 223 |
1
⊢
(∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪
𝑛 ∈ 𝑍 𝐴 ⊆ ∪
𝑛 ∈ 𝑍 𝐵) |