Step | Hyp | Ref
| Expression |
1 | | suctr 6296 |
. . . 4
⊢ (Tr 𝑥 → Tr suc 𝑥) |
2 | | vex 3412 |
. . . . 5
⊢ 𝑥 ∈ V |
3 | 2 | sucid 6292 |
. . . 4
⊢ 𝑥 ∈ suc 𝑥 |
4 | 2 | sucex 7590 |
. . . . 5
⊢ suc 𝑥 ∈ V |
5 | | treq 5167 |
. . . . . 6
⊢ (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥)) |
6 | | eleq2 2826 |
. . . . . 6
⊢ (𝑐 = suc 𝑥 → (𝑥 ∈ 𝑐 ↔ 𝑥 ∈ suc 𝑥)) |
7 | 5, 6 | anbi12d 634 |
. . . . 5
⊢ (𝑐 = suc 𝑥 → ((Tr 𝑐 ∧ 𝑥 ∈ 𝑐) ↔ (Tr suc 𝑥 ∧ 𝑥 ∈ suc 𝑥))) |
8 | 4, 7 | spcev 3521 |
. . . 4
⊢ ((Tr suc
𝑥 ∧ 𝑥 ∈ suc 𝑥) → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐)) |
9 | 1, 3, 8 | sylancl 589 |
. . 3
⊢ (Tr 𝑥 → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐)) |
10 | 9 | adantr 484 |
. 2
⊢ ((Tr
𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐)) |
11 | | simprl 771 |
. . . . . 6
⊢
((∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Tr 𝑎) |
12 | | dford3lem1 40551 |
. . . . . . . . 9
⊢ ((Tr
𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → ∀𝑏 ∈ 𝑎 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦)) |
13 | | ralim 3085 |
. . . . . . . . 9
⊢
(∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) → (∀𝑏 ∈ 𝑎 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → ∀𝑏 ∈ 𝑎 𝑏 ∈ On)) |
14 | 12, 13 | syl5 34 |
. . . . . . . 8
⊢
(∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → ∀𝑏 ∈ 𝑎 𝑏 ∈ On)) |
15 | 14 | imp 410 |
. . . . . . 7
⊢
((∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → ∀𝑏 ∈ 𝑎 𝑏 ∈ On) |
16 | | dfss3 3888 |
. . . . . . 7
⊢ (𝑎 ⊆ On ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ On) |
17 | 15, 16 | sylibr 237 |
. . . . . 6
⊢
((∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ⊆ On) |
18 | | ordon 7561 |
. . . . . . 7
⊢ Ord
On |
19 | 18 | a1i 11 |
. . . . . 6
⊢
((∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Ord On) |
20 | | trssord 6230 |
. . . . . 6
⊢ ((Tr
𝑎 ∧ 𝑎 ⊆ On ∧ Ord On) → Ord 𝑎) |
21 | 11, 17, 19, 20 | syl3anc 1373 |
. . . . 5
⊢
((∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → Ord 𝑎) |
22 | | vex 3412 |
. . . . . 6
⊢ 𝑎 ∈ V |
23 | 22 | elon 6222 |
. . . . 5
⊢ (𝑎 ∈ On ↔ Ord 𝑎) |
24 | 21, 23 | sylibr 237 |
. . . 4
⊢
((∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ∈ On) |
25 | 24 | ex 416 |
. . 3
⊢
(∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On)) |
26 | | treq 5167 |
. . . . 5
⊢ (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏)) |
27 | | raleq 3319 |
. . . . 5
⊢ (𝑎 = 𝑏 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑏 Tr 𝑦)) |
28 | 26, 27 | anbi12d 634 |
. . . 4
⊢ (𝑎 = 𝑏 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))) |
29 | | eleq1w 2820 |
. . . 4
⊢ (𝑎 = 𝑏 → (𝑎 ∈ On ↔ 𝑏 ∈ On)) |
30 | 28, 29 | imbi12d 348 |
. . 3
⊢ (𝑎 = 𝑏 → (((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On))) |
31 | | treq 5167 |
. . . . 5
⊢ (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥)) |
32 | | raleq 3319 |
. . . . 5
⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑥 Tr 𝑦)) |
33 | 31, 32 | anbi12d 634 |
. . . 4
⊢ (𝑎 = 𝑥 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦))) |
34 | | eleq1w 2820 |
. . . 4
⊢ (𝑎 = 𝑥 → (𝑎 ∈ On ↔ 𝑥 ∈ On)) |
35 | 33, 34 | imbi12d 348 |
. . 3
⊢ (𝑎 = 𝑥 → (((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On))) |
36 | 25, 30, 35 | setindtrs 40550 |
. 2
⊢
(∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐) → ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On)) |
37 | 10, 36 | mpcom 38 |
1
⊢ ((Tr
𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On) |