Step | Hyp | Ref
| Expression |
1 | | id 22 |
. . . . . 6
⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤) |
2 | 1 | cbvdisjv 5051 |
. . . . 5
⊢
(Disj 𝑦
∈ ran 𝐹 𝑦 ↔ Disj 𝑤 ∈ ran 𝐹 𝑤) |
3 | | id 22 |
. . . . . . 7
⊢ (𝑤 = 𝑣 → 𝑤 = 𝑣) |
4 | 3 | ndisj2 42606 |
. . . . . 6
⊢ (¬
Disj 𝑤 ∈ ran
𝐹 𝑤 ↔ ∃𝑤 ∈ ran 𝐹∃𝑣 ∈ ran 𝐹(𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) |
5 | 4 | biimpi 215 |
. . . . 5
⊢ (¬
Disj 𝑤 ∈ ran
𝐹 𝑤 → ∃𝑤 ∈ ran 𝐹∃𝑣 ∈ ran 𝐹(𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) |
6 | 2, 5 | sylnbi 330 |
. . . 4
⊢ (¬
Disj 𝑦 ∈ ran
𝐹 𝑦 → ∃𝑤 ∈ ran 𝐹∃𝑣 ∈ ran 𝐹(𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) |
7 | | disjrnmpt2.1 |
. . . . . . . . . . . . 13
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
8 | 7 | elrnmpt 5868 |
. . . . . . . . . . . 12
⊢ (𝑤 ∈ ran 𝐹 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)) |
9 | 8 | ibi 266 |
. . . . . . . . . . 11
⊢ (𝑤 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 𝑤 = 𝐵) |
10 | | nfcv 2908 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑧𝐵 |
11 | | nfcsb1v 3858 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵 |
12 | | csbeq1a 3847 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
13 | 10, 11, 12 | cbvmpt 5186 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝐵) |
14 | 7, 13 | eqtri 2767 |
. . . . . . . . . . . . 13
⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝐵) |
15 | 14 | elrnmpt 5868 |
. . . . . . . . . . . 12
⊢ (𝑣 ∈ ran 𝐹 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) |
16 | 15 | ibi 266 |
. . . . . . . . . . 11
⊢ (𝑣 ∈ ran 𝐹 → ∃𝑧 ∈ 𝐴 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) |
17 | 9, 16 | anim12i 613 |
. . . . . . . . . 10
⊢ ((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) → (∃𝑥 ∈ 𝐴 𝑤 = 𝐵 ∧ ∃𝑧 ∈ 𝐴 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) |
18 | | nfv 1918 |
. . . . . . . . . . 11
⊢
Ⅎ𝑧 𝑤 = 𝐵 |
19 | 11 | nfeq2 2925 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌𝐵 |
20 | 18, 19 | reean 3294 |
. . . . . . . . . 10
⊢
(∃𝑥 ∈
𝐴 ∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) ↔ (∃𝑥 ∈ 𝐴 𝑤 = 𝐵 ∧ ∃𝑧 ∈ 𝐴 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) |
21 | 17, 20 | sylibr 233 |
. . . . . . . . 9
⊢ ((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) |
22 | 21 | adantr 481 |
. . . . . . . 8
⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) |
23 | | nfmpt1 5183 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) |
24 | 7, 23 | nfcxfr 2906 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥𝐹 |
25 | 24 | nfrn 5864 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥ran
𝐹 |
26 | 25 | nfcri 2895 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑤 ∈ ran 𝐹 |
27 | 25 | nfcri 2895 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑣 ∈ ran 𝐹 |
28 | 26, 27 | nfan 1903 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) |
29 | | nfv 1918 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) |
30 | 28, 29 | nfan 1903 |
. . . . . . . . 9
⊢
Ⅎ𝑥((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) |
31 | | simpll 764 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑥 = 𝑧) → 𝑤 = 𝐵) |
32 | 12 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑥 = 𝑧) → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
33 | | id 22 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 = ⦋𝑧 / 𝑥⦌𝐵 → 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) |
34 | 33 | eqcomd 2745 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 = ⦋𝑧 / 𝑥⦌𝐵 → ⦋𝑧 / 𝑥⦌𝐵 = 𝑣) |
35 | 34 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑥 = 𝑧) → ⦋𝑧 / 𝑥⦌𝐵 = 𝑣) |
36 | 31, 32, 35 | 3eqtrd 2783 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑥 = 𝑧) → 𝑤 = 𝑣) |
37 | 36 | adantll 711 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) ∧ 𝑥 = 𝑧) → 𝑤 = 𝑣) |
38 | | simpll 764 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) ∧ 𝑥 = 𝑧) → 𝑤 ≠ 𝑣) |
39 | 38 | neneqd 2949 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) ∧ 𝑥 = 𝑧) → ¬ 𝑤 = 𝑣) |
40 | 37, 39 | pm2.65da 814 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑤 ≠ 𝑣 ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → ¬ 𝑥 = 𝑧) |
41 | 40 | neqned 2951 |
. . . . . . . . . . . . . . 15
⊢ ((𝑤 ≠ 𝑣 ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → 𝑥 ≠ 𝑧) |
42 | 41 | adantlr 712 |
. . . . . . . . . . . . . 14
⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → 𝑥 ≠ 𝑧) |
43 | | id 22 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = 𝐵 → 𝑤 = 𝐵) |
44 | 43 | eqcomd 2745 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 = 𝐵 → 𝐵 = 𝑤) |
45 | 44 | ad2antrl 725 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑤 ∩ 𝑣) ≠ ∅ ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → 𝐵 = 𝑤) |
46 | 34 | ad2antll 726 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑤 ∩ 𝑣) ≠ ∅ ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → ⦋𝑧 / 𝑥⦌𝐵 = 𝑣) |
47 | 45, 46 | ineq12d 4148 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑤 ∩ 𝑣) ≠ ∅ ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = (𝑤 ∩ 𝑣)) |
48 | | simpl 483 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑤 ∩ 𝑣) ≠ ∅ ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → (𝑤 ∩ 𝑣) ≠ ∅) |
49 | 47, 48 | eqnetrd 3012 |
. . . . . . . . . . . . . . 15
⊢ (((𝑤 ∩ 𝑣) ≠ ∅ ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅) |
50 | 49 | adantll 711 |
. . . . . . . . . . . . . 14
⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅) |
51 | 42, 50 | jca 512 |
. . . . . . . . . . . . 13
⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)) |
52 | 51 | ex 413 |
. . . . . . . . . . . 12
⊢ ((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) → ((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) → (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))) |
53 | 52 | adantl 482 |
. . . . . . . . . . 11
⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → ((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) → (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))) |
54 | 53 | reximdv 3203 |
. . . . . . . . . 10
⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → (∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) → ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))) |
55 | 54 | a1d 25 |
. . . . . . . . 9
⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → (𝑥 ∈ 𝐴 → (∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) → ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)))) |
56 | 30, 55 | reximdai 3245 |
. . . . . . . 8
⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))) |
57 | 22, 56 | mpd 15 |
. . . . . . 7
⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)) |
58 | 57 | ex 413 |
. . . . . 6
⊢ ((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) → ((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))) |
59 | 58 | a1i 11 |
. . . . 5
⊢ (¬
Disj 𝑦 ∈ ran
𝐹 𝑦 → ((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) → ((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)))) |
60 | 59 | rexlimdvv 3223 |
. . . 4
⊢ (¬
Disj 𝑦 ∈ ran
𝐹 𝑦 → (∃𝑤 ∈ ran 𝐹∃𝑣 ∈ ran 𝐹(𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))) |
61 | 6, 60 | mpd 15 |
. . 3
⊢ (¬
Disj 𝑦 ∈ ran
𝐹 𝑦 → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)) |
62 | | csbeq1 3836 |
. . . . . 6
⊢ (𝑢 = 𝑧 → ⦋𝑢 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵) |
63 | 62 | ndisj2 42606 |
. . . . 5
⊢ (¬
Disj 𝑢 ∈ 𝐴 ⦋𝑢 / 𝑥⦌𝐵 ↔ ∃𝑢 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)) |
64 | | nfcv 2908 |
. . . . . . 7
⊢
Ⅎ𝑥𝐴 |
65 | | nfv 1918 |
. . . . . . . 8
⊢
Ⅎ𝑥 𝑢 ≠ 𝑧 |
66 | | nfcsb1v 3858 |
. . . . . . . . . 10
⊢
Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵 |
67 | 66, 11 | nfin 4151 |
. . . . . . . . 9
⊢
Ⅎ𝑥(⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) |
68 | | nfcv 2908 |
. . . . . . . . 9
⊢
Ⅎ𝑥∅ |
69 | 67, 68 | nfne 3046 |
. . . . . . . 8
⊢
Ⅎ𝑥(⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅ |
70 | 65, 69 | nfan 1903 |
. . . . . . 7
⊢
Ⅎ𝑥(𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅) |
71 | 64, 70 | nfrex 3243 |
. . . . . 6
⊢
Ⅎ𝑥∃𝑧 ∈ 𝐴 (𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅) |
72 | | nfv 1918 |
. . . . . 6
⊢
Ⅎ𝑢∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅) |
73 | | neeq1 3007 |
. . . . . . . 8
⊢ (𝑢 = 𝑥 → (𝑢 ≠ 𝑧 ↔ 𝑥 ≠ 𝑧)) |
74 | | csbeq1 3836 |
. . . . . . . . . . 11
⊢ (𝑢 = 𝑥 → ⦋𝑢 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵) |
75 | | csbid 3846 |
. . . . . . . . . . 11
⊢
⦋𝑥 /
𝑥⦌𝐵 = 𝐵 |
76 | 74, 75 | eqtrdi 2795 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑥 → ⦋𝑢 / 𝑥⦌𝐵 = 𝐵) |
77 | 76 | ineq1d 4146 |
. . . . . . . . 9
⊢ (𝑢 = 𝑥 → (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) |
78 | 77 | neeq1d 3004 |
. . . . . . . 8
⊢ (𝑢 = 𝑥 → ((⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅ ↔ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)) |
79 | 73, 78 | anbi12d 631 |
. . . . . . 7
⊢ (𝑢 = 𝑥 → ((𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅) ↔ (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))) |
80 | 79 | rexbidv 3227 |
. . . . . 6
⊢ (𝑢 = 𝑥 → (∃𝑧 ∈ 𝐴 (𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅) ↔ ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))) |
81 | 71, 72, 80 | cbvrexw 3375 |
. . . . 5
⊢
(∃𝑢 ∈
𝐴 ∃𝑧 ∈ 𝐴 (𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)) |
82 | 63, 81 | bitri 274 |
. . . 4
⊢ (¬
Disj 𝑢 ∈ 𝐴 ⦋𝑢 / 𝑥⦌𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)) |
83 | | nfcv 2908 |
. . . . 5
⊢
Ⅎ𝑢𝐵 |
84 | | csbeq1a 3847 |
. . . . 5
⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵) |
85 | 83, 66, 84 | cbvdisj 5050 |
. . . 4
⊢
(Disj 𝑥
∈ 𝐴 𝐵 ↔ Disj 𝑢 ∈ 𝐴 ⦋𝑢 / 𝑥⦌𝐵) |
86 | 82, 85 | xchnxbir 333 |
. . 3
⊢ (¬
Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)) |
87 | 61, 86 | sylibr 233 |
. 2
⊢ (¬
Disj 𝑦 ∈ ran
𝐹 𝑦 → ¬ Disj 𝑥 ∈ 𝐴 𝐵) |
88 | 87 | con4i 114 |
1
⊢
(Disj 𝑥
∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran 𝐹 𝑦) |