| Step | Hyp | Ref
| Expression |
| 1 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑚𝐴 |
| 2 | | nfcsb1v 3923 |
. . . . . . . . 9
⊢
Ⅎ𝑛⦋𝑚 / 𝑛⦌𝐴 |
| 3 | | csbeq1a 3913 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → 𝐴 = ⦋𝑚 / 𝑛⦌𝐴) |
| 4 | 1, 2, 3 | cbviun 5036 |
. . . . . . . 8
⊢ ∪ 𝑛 ∈ 𝑍 𝐴 = ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴 |
| 5 | 4 | eleq2i 2833 |
. . . . . . 7
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ 𝑥 ∈ ∪
𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴) |
| 6 | | eliun 4995 |
. . . . . . 7
⊢ (𝑥 ∈ ∪ 𝑚 ∈ 𝑍 ⦋𝑚 / 𝑛⦌𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) |
| 7 | 5, 6 | bitri 275 |
. . . . . 6
⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐴 ↔ ∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) |
| 8 | 7 | biimpi 216 |
. . . . 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 3284 |
. . . . . 6
⊢
Ⅎ𝑚∀𝑚 ∈ 𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 |
| 11 | | nfv 1914 |
. . . . . 6
⊢
Ⅎ𝑚 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 𝐵 |
| 12 | | simp2 1138 |
. . . . . . . 8
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑚 ∈ 𝑍) |
| 13 | | rspa 3248 |
. . . . . . . . 9
⊢
((∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑚 ∈ 𝑍) → ∪
𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵) |
| 14 | 13 | 3adant3 1133 |
. . . . . . . 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 1139 |
. . . . . . . 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 13605 |
. . . . . . . . . . . . 13
⊢ (𝑁...𝑚) ⊆ (ℤ≥‘𝑁) |
| 18 | | iuneqfzuzlem.z |
. . . . . . . . . . . . . 14
⊢ 𝑍 =
(ℤ≥‘𝑁) |
| 19 | 18 | eqcomi 2746 |
. . . . . . . . . . . . 13
⊢
(ℤ≥‘𝑁) = 𝑍 |
| 20 | 17, 19 | sseqtri 4032 |
. . . . . . . . . . . 12
⊢ (𝑁...𝑚) ⊆ 𝑍 |
| 21 | | iunss1 5006 |
. . . . . . . . . . . 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 4018 |
. . . . . . . . . 10
⊢ (∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪
𝑛 ∈ (𝑁...𝑚)𝐴 ⊆ ∪
𝑛 ∈ 𝑍 𝐵) |
| 24 | 23 | 3ad2ant2 1135 |
. . . . . . . . 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 2833 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ 𝑍 ↔ 𝑚 ∈ (ℤ≥‘𝑁)) |
| 26 | 25 | biimpi 216 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (ℤ≥‘𝑁)) |
| 27 | | eluzel2 12883 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈
(ℤ≥‘𝑁) → 𝑁 ∈ ℤ) |
| 28 | 26, 27 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ 𝑍 → 𝑁 ∈ ℤ) |
| 29 | | eluzelz 12888 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈
(ℤ≥‘𝑁) → 𝑚 ∈ ℤ) |
| 30 | 26, 29 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℤ) |
| 31 | | eluzle 12891 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈
(ℤ≥‘𝑁) → 𝑁 ≤ 𝑚) |
| 32 | 26, 31 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ 𝑍 → 𝑁 ≤ 𝑚) |
| 33 | 30 | zred 12722 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ ℝ) |
| 34 | | leid 11357 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℝ → 𝑚 ≤ 𝑚) |
| 35 | 33, 34 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ 𝑍 → 𝑚 ≤ 𝑚) |
| 36 | 28, 30, 30, 32, 35 | elfzd 13555 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ (𝑁...𝑚)) |
| 37 | | nfcv 2905 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑛𝑥 |
| 38 | 37, 2 | nfel 2920 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑛 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 |
| 39 | 3 | eleq2d 2827 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴)) |
| 40 | 38, 39 | rspce 3611 |
. . . . . . . . . . . 12
⊢ ((𝑚 ∈ (𝑁...𝑚) ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴) |
| 41 | 36, 40 | sylan 580 |
. . . . . . . . . . 11
⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴) |
| 42 | | eliun 4995 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 ↔ ∃𝑛 ∈ (𝑁...𝑚)𝑥 ∈ 𝐴) |
| 43 | 41, 42 | sylibr 234 |
. . . . . . . . . 10
⊢ ((𝑚 ∈ 𝑍 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪
𝑛 ∈ (𝑁...𝑚)𝐴) |
| 44 | 43 | 3adant2 1132 |
. . . . . . . . 9
⊢ ((𝑚 ∈ 𝑍 ∧ ∪
𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 ∧ 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴) → 𝑥 ∈ ∪
𝑛 ∈ (𝑁...𝑚)𝐴) |
| 45 | 24, 44 | sseldd 3984 |
. . . . . . . 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 1120 |
. . . . . 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 3266 |
. . . . 5
⊢
(∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → (∃𝑚 ∈ 𝑍 𝑥 ∈ ⦋𝑚 / 𝑛⦌𝐴 → 𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵)) |
| 49 | 48 | 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)) |
| 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 3146 |
. 2
⊢
(∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∀𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
| 52 | | dfss3 3972 |
. 2
⊢ (∪ 𝑛 ∈ 𝑍 𝐴 ⊆ ∪
𝑛 ∈ 𝑍 𝐵 ↔ ∀𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐴𝑥 ∈ ∪
𝑛 ∈ 𝑍 𝐵) |
| 53 | 51, 52 | sylibr 234 |
1
⊢
(∀𝑚 ∈
𝑍 ∪ 𝑛 ∈ (𝑁...𝑚)𝐴 = ∪ 𝑛 ∈ (𝑁...𝑚)𝐵 → ∪
𝑛 ∈ 𝑍 𝐴 ⊆ ∪
𝑛 ∈ 𝑍 𝐵) |