Step | Hyp | Ref
| Expression |
1 | | disjel 4395 |
. . . . . . . . . . 11
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ 𝐵) |
2 | | eleq1w 2822 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) |
3 | 2 | notbid 317 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝐵 ↔ ¬ 𝑦 ∈ 𝐵)) |
4 | 1, 3 | syl5ibcom 244 |
. . . . . . . . . 10
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝑦 → ¬ 𝑦 ∈ 𝐵)) |
5 | 4 | con2d 134 |
. . . . . . . . 9
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 → ¬ 𝑥 = 𝑦)) |
6 | 5 | impr 454 |
. . . . . . . 8
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ 𝑥 = 𝑦) |
7 | | biorf 933 |
. . . . . . . 8
⊢ (¬
𝑥 = 𝑦 → ((𝐶 ∩ 𝐷) = ∅ ↔ (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))) |
8 | 6, 7 | syl 17 |
. . . . . . 7
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝐶 ∩ 𝐷) = ∅ ↔ (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))) |
9 | 8 | bicomd 222 |
. . . . . 6
⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ (𝐶 ∩ 𝐷) = ∅)) |
10 | 9 | 2ralbidva 3123 |
. . . . 5
⊢ ((𝐴 ∩ 𝐵) = ∅ → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)) |
11 | 10 | anbi2d 628 |
. . . 4
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅))) |
12 | | ralunb 4129 |
. . . . . 6
⊢
(∀𝑦 ∈
(𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ (∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))) |
13 | 12 | ralbii 3092 |
. . . . 5
⊢
(∀𝑥 ∈
𝐴 ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))) |
14 | | nfv 1920 |
. . . . . 6
⊢
Ⅎ𝑧∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) |
15 | | nfcv 2908 |
. . . . . . 7
⊢
Ⅎ𝑥(𝐴 ∪ 𝐵) |
16 | | nfv 1920 |
. . . . . . . 8
⊢
Ⅎ𝑥 𝑧 = 𝑤 |
17 | | nfcsb1v 3861 |
. . . . . . . . . 10
⊢
Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐶 |
18 | | nfcsb1v 3861 |
. . . . . . . . . 10
⊢
Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐶 |
19 | 17, 18 | nfin 4155 |
. . . . . . . . 9
⊢
Ⅎ𝑥(⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) |
20 | 19 | nfeq1 2923 |
. . . . . . . 8
⊢
Ⅎ𝑥(⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅ |
21 | 16, 20 | nfor 1910 |
. . . . . . 7
⊢
Ⅎ𝑥(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) |
22 | 15, 21 | nfralw 3151 |
. . . . . 6
⊢
Ⅎ𝑥∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) |
23 | | equequ2 2032 |
. . . . . . . . 9
⊢ (𝑤 = 𝑦 → (𝑥 = 𝑤 ↔ 𝑥 = 𝑦)) |
24 | | nfcv 2908 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥𝑦 |
25 | | nfcv 2908 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥𝐷 |
26 | | disjxun.1 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
27 | 24, 25, 26 | csbhypf 3865 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑦 → ⦋𝑤 / 𝑥⦌𝐶 = 𝐷) |
28 | 27 | ineq2d 4151 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑦 → (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = (𝐶 ∩ 𝐷)) |
29 | 28 | eqeq1d 2741 |
. . . . . . . . 9
⊢ (𝑤 = 𝑦 → ((𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅ ↔ (𝐶 ∩ 𝐷) = ∅)) |
30 | 23, 29 | orbi12d 915 |
. . . . . . . 8
⊢ (𝑤 = 𝑦 → ((𝑥 = 𝑤 ∨ (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))) |
31 | 30 | cbvralvw 3380 |
. . . . . . 7
⊢
(∀𝑤 ∈
(𝐴 ∪ 𝐵)(𝑥 = 𝑤 ∨ (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)) |
32 | | equequ1 2031 |
. . . . . . . . 9
⊢ (𝑥 = 𝑧 → (𝑥 = 𝑤 ↔ 𝑧 = 𝑤)) |
33 | | csbeq1a 3850 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑧 → 𝐶 = ⦋𝑧 / 𝑥⦌𝐶) |
34 | 33 | ineq1d 4150 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑧 → (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶)) |
35 | 34 | eqeq1d 2741 |
. . . . . . . . 9
⊢ (𝑥 = 𝑧 → ((𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅ ↔ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) |
36 | 32, 35 | orbi12d 915 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → ((𝑥 = 𝑤 ∨ (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))) |
37 | 36 | ralbidv 3122 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → (∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑤 ∨ (𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))) |
38 | 31, 37 | bitr3id 284 |
. . . . . 6
⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))) |
39 | 14, 22, 38 | cbvralw 3371 |
. . . . 5
⊢
(∀𝑥 ∈
𝐴 ∀𝑦 ∈ (𝐴 ∪ 𝐵)(𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) |
40 | | r19.26 3096 |
. . . . 5
⊢
(∀𝑥 ∈
𝐴 (∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))) |
41 | 13, 39, 40 | 3bitr3i 300 |
. . . 4
⊢
(∀𝑧 ∈
𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))) |
42 | 26 | disjor 5058 |
. . . . 5
⊢
(Disj 𝑥
∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)) |
43 | 42 | anbi1i 623 |
. . . 4
⊢
((Disj 𝑥
∈ 𝐴 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)) |
44 | 11, 41, 43 | 3bitr4g 313 |
. . 3
⊢ ((𝐴 ∩ 𝐵) = ∅ → (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (Disj 𝑥 ∈ 𝐴 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅))) |
45 | | nfv 1920 |
. . . . . . . . . 10
⊢
Ⅎ𝑤(𝑧 = 𝑥 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅) |
46 | | equequ2 2032 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑤 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑤)) |
47 | | csbeq1a 3850 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑤 → 𝐶 = ⦋𝑤 / 𝑥⦌𝐶) |
48 | 47 | ineq2d 4151 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑤 → (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶)) |
49 | 48 | eqeq1d 2741 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑤 → ((⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅ ↔ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) |
50 | 46, 49 | orbi12d 915 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑤 → ((𝑧 = 𝑥 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅) ↔ (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))) |
51 | 45, 21, 50 | cbvralw 3371 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
𝐴 (𝑧 = 𝑥 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅) ↔ ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) |
52 | | equequ1 2031 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑦 = 𝑥)) |
53 | | equcom 2024 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) |
54 | 52, 53 | bitrdi 286 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑦 → (𝑧 = 𝑥 ↔ 𝑥 = 𝑦)) |
55 | 24, 25, 26 | csbhypf 3865 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑥⦌𝐶 = 𝐷) |
56 | 55 | ineq1d 4150 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑦 → (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = (𝐷 ∩ 𝐶)) |
57 | | incom 4139 |
. . . . . . . . . . . . 13
⊢ (𝐷 ∩ 𝐶) = (𝐶 ∩ 𝐷) |
58 | 56, 57 | eqtrdi 2795 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑦 → (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = (𝐶 ∩ 𝐷)) |
59 | 58 | eqeq1d 2741 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑦 → ((⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅ ↔ (𝐶 ∩ 𝐷) = ∅)) |
60 | 54, 59 | orbi12d 915 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑦 → ((𝑧 = 𝑥 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅) ↔ (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))) |
61 | 60 | ralbidv 3122 |
. . . . . . . . 9
⊢ (𝑧 = 𝑦 → (∀𝑥 ∈ 𝐴 (𝑧 = 𝑥 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ 𝐶) = ∅) ↔ ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))) |
62 | 51, 61 | bitr3id 284 |
. . . . . . . 8
⊢ (𝑧 = 𝑦 → (∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅))) |
63 | 62 | cbvralvw 3380 |
. . . . . . 7
⊢
(∀𝑧 ∈
𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)) |
64 | | ralcom 3282 |
. . . . . . 7
⊢
(∀𝑦 ∈
𝐵 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)) |
65 | 63, 64 | bitri 274 |
. . . . . 6
⊢
(∀𝑧 ∈
𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥 = 𝑦 ∨ (𝐶 ∩ 𝐷) = ∅)) |
66 | 65, 10 | syl5bb 282 |
. . . . 5
⊢ ((𝐴 ∩ 𝐵) = ∅ → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)) |
67 | 66 | anbi1d 629 |
. . . 4
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅ ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)))) |
68 | | ralunb 4129 |
. . . . . 6
⊢
(∀𝑤 ∈
(𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))) |
69 | 68 | ralbii 3092 |
. . . . 5
⊢
(∀𝑧 ∈
𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ ∀𝑧 ∈ 𝐵 (∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))) |
70 | | r19.26 3096 |
. . . . 5
⊢
(∀𝑧 ∈
𝐵 (∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) ↔ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))) |
71 | 69, 70 | bitri 274 |
. . . 4
⊢
(∀𝑧 ∈
𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐴 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))) |
72 | | disjors 5059 |
. . . . 5
⊢
(Disj 𝑥
∈ 𝐵 𝐶 ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) |
73 | 72 | anbi2ci 624 |
. . . 4
⊢
((Disj 𝑥
∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅ ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))) |
74 | 67, 71, 73 | 3bitr4g 313 |
. . 3
⊢ ((𝐴 ∩ 𝐵) = ∅ → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅))) |
75 | 44, 74 | anbi12d 630 |
. 2
⊢ ((𝐴 ∩ 𝐵) = ∅ → ((∀𝑧 ∈ 𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) ↔ ((Disj 𝑥 ∈ 𝐴 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ∧ (Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)))) |
76 | | disjors 5059 |
. . 3
⊢
(Disj 𝑥
∈ (𝐴 ∪ 𝐵)𝐶 ↔ ∀𝑧 ∈ (𝐴 ∪ 𝐵)∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅)) |
77 | | ralunb 4129 |
. . 3
⊢
(∀𝑧 ∈
(𝐴 ∪ 𝐵)∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ↔ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))) |
78 | 76, 77 | bitri 274 |
. 2
⊢
(Disj 𝑥
∈ (𝐴 ∪ 𝐵)𝐶 ↔ (∀𝑧 ∈ 𝐴 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ (𝐴 ∪ 𝐵)(𝑧 = 𝑤 ∨ (⦋𝑧 / 𝑥⦌𝐶 ∩ ⦋𝑤 / 𝑥⦌𝐶) = ∅))) |
79 | | df-3an 1087 |
. . 3
⊢
((Disj 𝑥
∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ↔ ((Disj 𝑥 ∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅)) |
80 | | anandir 673 |
. . 3
⊢
(((Disj 𝑥
∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ↔ ((Disj 𝑥 ∈ 𝐴 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ∧ (Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅))) |
81 | 79, 80 | bitri 274 |
. 2
⊢
((Disj 𝑥
∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ↔ ((Disj 𝑥 ∈ 𝐴 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅) ∧ (Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅))) |
82 | 75, 78, 81 | 3bitr4g 313 |
1
⊢ ((𝐴 ∩ 𝐵) = ∅ → (Disj 𝑥 ∈ (𝐴 ∪ 𝐵)𝐶 ↔ (Disj 𝑥 ∈ 𝐴 𝐶 ∧ Disj 𝑥 ∈ 𝐵 𝐶 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∩ 𝐷) = ∅))) |