Step | Hyp | Ref
| Expression |
1 | | nfcv 2904 |
. . . . . . . . 9
⊢
Ⅎ𝑚𝐴 |
2 | | nfcsb1v 3836 |
. . . . . . . . 9
⊢
Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 |
3 | | csbeq1a 3825 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴) |
4 | 1, 2, 3 | cbviun 4945 |
. . . . . . . 8
⊢ ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴 |
5 | 4 | eleq2i 2829 |
. . . . . . 7
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ 𝑥 ∈ ∪
𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴) |
6 | | eliun 4908 |
. . . . . . 7
⊢ (𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) |
7 | 5, 6 | bitri 278 |
. . . . . 6
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) |
8 | 7 | biimpi 219 |
. . . . 5
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 → ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) |
9 | 8 | adantl 485 |
. . . 4
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐴) → ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) |
10 | | nfra1 3140 |
. . . . . 6
⊢
Ⅎ𝑚∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 |
11 | | nfv 1922 |
. . . . . 6
⊢
Ⅎ𝑚 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵 |
12 | | simp2 1139 |
. . . . . . . 8
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑚 ∈ 𝑍) |
13 | | rspa 3128 |
. . . . . . . . 9
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍) → ∪
𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵) |
14 | 13 | 3adant3 1134 |
. . . . . . . 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 1140 |
. . . . . . . 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 13153 |
. . . . . . . . . . . . 13
⊢ (𝑁...𝑚) ⊆ (ℤ≥‘𝑁) |
18 | | iuneqfzuzlem.z |
. . . . . . . . . . . . . 14
⊢ 𝑍 =
(ℤ≥‘𝑁) |
19 | 18 | eqcomi 2746 |
. . . . . . . . . . . . 13
⊢
(ℤ≥‘𝑁) = 𝑍 |
20 | 17, 19 | sseqtri 3937 |
. . . . . . . . . . . 12
⊢ (𝑁...𝑚) ⊆ 𝑍 |
21 | | iunss1 4918 |
. . . . . . . . . . . 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 3939 |
. . . . . . . . . 10
⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪
𝑛 ∈ (𝑁...𝑚)𝐴 ⊆ ∪
𝑛 ∈ 𝑍 𝐵) |
24 | 23 | 3ad2ant2 1136 |
. . . . . . . . 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 2829 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ 𝑍 ↔ 𝑚 ∈ (ℤ≥‘𝑁)) |
26 | 25 | biimpi 219 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (ℤ≥‘𝑁)) |
27 | | eluzel2 12443 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈
(ℤ≥‘𝑁) → 𝑁 ∈ ℤ) |
28 | 26, 27 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ 𝑍 → 𝑁 ∈ ℤ) |
29 | | eluzelz 12448 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈
(ℤ≥‘𝑁) → 𝑚 ∈ ℤ) |
30 | 26, 29 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ) |
31 | | eluzle 12451 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈
(ℤ≥‘𝑁) → 𝑁 ≤ 𝑚) |
32 | 26, 31 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ 𝑍 → 𝑁 ≤ 𝑚) |
33 | 30 | zred 12282 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℝ) |
34 | | leid 10928 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℝ → 𝑚 ≤ 𝑚) |
35 | 33, 34 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ 𝑍 → 𝑚 ≤ 𝑚) |
36 | 28, 30, 30, 32, 35 | elfzd 13103 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (𝑁...𝑚)) |
37 | | nfcv 2904 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛𝑥 |
38 | 37, 2 | nfel 2918 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 |
39 | 3 | eleq2d 2823 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)) |
40 | 38, 39 | rspce 3526 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ (𝑁...𝑚) ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴) |
41 | 36, 40 | sylan 583 |
. . . . . . . . . . 11
⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴) |
42 | | eliun 4908 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 ↔ ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴) |
43 | 41, 42 | sylibr 237 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪
𝑛 ∈ (𝑁...𝑚)𝐴) |
44 | 43 | 3adant2 1133 |
. . . . . . . . 9
⊢ ((𝑚 ∈ 𝑍 ∧ ∪
𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪
𝑛 ∈ (𝑁...𝑚)𝐴) |
45 | 24, 44 | sseldd 3902 |
. . . . . . . 8
⊢ ((𝑚 ∈ 𝑍 ∧ ∪
𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
46 | 12, 14, 15, 45 | syl3anc 1373 |
. . . . . . 7
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
47 | 46 | 3exp 1121 |
. . . . . 6
⊢
(∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → (𝑚 ∈ 𝑍 → (𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵))) |
48 | 10, 11, 47 | rexlimd 3236 |
. . . . 5
⊢
(∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → (∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵)) |
49 | 48 | adantr 484 |
. . . 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)) |
50 | 9, 49 | mpd 15 |
. . 3
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐴) → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
51 | 50 | ralrimiva 3105 |
. 2
⊢
(∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
52 | | dfss3 3888 |
. 2
⊢ (∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪
𝑛 ∈ 𝑍 𝐵 ↔ ∀𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
53 | 51, 52 | sylibr 237 |
1
⊢
(∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪
𝑛 ∈ 𝑍 𝐴 ⊆ ∪
𝑛 ∈ 𝑍 𝐵) |