| Step | Hyp | Ref
| Expression |
| 1 | | suctr 6470 |
. . . 4
⊢ (Tr 𝑥 → Tr suc 𝑥) |
| 2 | | vex 3484 |
. . . . 5
⊢ 𝑥 ∈ V |
| 3 | 2 | sucid 6466 |
. . . 4
⊢ 𝑥 ∈ suc 𝑥 |
| 4 | 2 | sucex 7826 |
. . . . 5
⊢ suc 𝑥 ∈ V |
| 5 | | treq 5267 |
. . . . . 6
⊢ (𝑐 = suc 𝑥 → (Tr 𝑐 ↔ Tr suc 𝑥)) |
| 6 | | eleq2 2830 |
. . . . . 6
⊢ (𝑐 = suc 𝑥 → (𝑥 ∈ 𝑐 ↔ 𝑥 ∈ suc 𝑥)) |
| 7 | 5, 6 | anbi12d 632 |
. . . . 5
⊢ (𝑐 = suc 𝑥 → ((Tr 𝑐 ∧ 𝑥 ∈ 𝑐) ↔ (Tr suc 𝑥 ∧ 𝑥 ∈ suc 𝑥))) |
| 8 | 4, 7 | spcev 3606 |
. . . 4
⊢ ((Tr suc
𝑥 ∧ 𝑥 ∈ suc 𝑥) → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐)) |
| 9 | 1, 3, 8 | sylancl 586 |
. . 3
⊢ (Tr 𝑥 → ∃𝑐(Tr 𝑐 ∧ 𝑥 ∈ 𝑐)) |
| 10 | 9 | adantr 480 |
. 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 43038 |
. . . . . . . . 9
⊢ ((Tr
𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → ∀𝑏 ∈ 𝑎 (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦)) |
| 13 | | ralim 3086 |
. . . . . . . . 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 406 |
. . . . . . 7
⊢
((∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → ∀𝑏 ∈ 𝑎 𝑏 ∈ On) |
| 16 | | dfss3 3972 |
. . . . . . 7
⊢ (𝑎 ⊆ On ↔ ∀𝑏 ∈ 𝑎 𝑏 ∈ On) |
| 17 | 15, 16 | sylibr 234 |
. . . . . 6
⊢
((∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ⊆ On) |
| 18 | | ordon 7797 |
. . . . . . 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 6401 |
. . . . . 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 3484 |
. . . . . 6
⊢ 𝑎 ∈ V |
| 23 | 22 | elon 6393 |
. . . . 5
⊢ (𝑎 ∈ On ↔ Ord 𝑎) |
| 24 | 21, 23 | sylibr 234 |
. . . 4
⊢
((∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) ∧ (Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦)) → 𝑎 ∈ On) |
| 25 | 24 | ex 412 |
. . 3
⊢
(∀𝑏 ∈
𝑎 ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On) → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On)) |
| 26 | | treq 5267 |
. . . . 5
⊢ (𝑎 = 𝑏 → (Tr 𝑎 ↔ Tr 𝑏)) |
| 27 | | raleq 3323 |
. . . . 5
⊢ (𝑎 = 𝑏 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑏 Tr 𝑦)) |
| 28 | 26, 27 | anbi12d 632 |
. . . 4
⊢ (𝑎 = 𝑏 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦))) |
| 29 | | eleq1w 2824 |
. . . 4
⊢ (𝑎 = 𝑏 → (𝑎 ∈ On ↔ 𝑏 ∈ On)) |
| 30 | 28, 29 | imbi12d 344 |
. . 3
⊢ (𝑎 = 𝑏 → (((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑏 ∧ ∀𝑦 ∈ 𝑏 Tr 𝑦) → 𝑏 ∈ On))) |
| 31 | | treq 5267 |
. . . . 5
⊢ (𝑎 = 𝑥 → (Tr 𝑎 ↔ Tr 𝑥)) |
| 32 | | raleq 3323 |
. . . . 5
⊢ (𝑎 = 𝑥 → (∀𝑦 ∈ 𝑎 Tr 𝑦 ↔ ∀𝑦 ∈ 𝑥 Tr 𝑦)) |
| 33 | 31, 32 | anbi12d 632 |
. . . 4
⊢ (𝑎 = 𝑥 → ((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) ↔ (Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦))) |
| 34 | | eleq1w 2824 |
. . . 4
⊢ (𝑎 = 𝑥 → (𝑎 ∈ On ↔ 𝑥 ∈ On)) |
| 35 | 33, 34 | imbi12d 344 |
. . 3
⊢ (𝑎 = 𝑥 → (((Tr 𝑎 ∧ ∀𝑦 ∈ 𝑎 Tr 𝑦) → 𝑎 ∈ On) ↔ ((Tr 𝑥 ∧ ∀𝑦 ∈ 𝑥 Tr 𝑦) → 𝑥 ∈ On))) |
| 36 | 25, 30, 35 | setindtrs 43037 |
. 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) |