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| Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.) | 
| Ref | Expression | 
|---|---|
| suctr | ⊢ (Tr 𝐴 → Tr suc 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elsuci 6451 | . . . . . 6 ⊢ (𝑦 ∈ suc 𝐴 → (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)) | |
| 2 | trel 5268 | . . . . . . . 8 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) | |
| 3 | 2 | expdimp 452 | . . . . . . 7 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∈ 𝐴 → 𝑧 ∈ 𝐴)) | 
| 4 | eleq2 2830 | . . . . . . . . 9 ⊢ (𝑦 = 𝐴 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝐴)) | |
| 5 | 4 | biimpcd 249 | . . . . . . . 8 ⊢ (𝑧 ∈ 𝑦 → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴)) | 
| 6 | 5 | adantl 481 | . . . . . . 7 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → (𝑦 = 𝐴 → 𝑧 ∈ 𝐴)) | 
| 7 | 3, 6 | jaod 860 | . . . . . 6 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → 𝑧 ∈ 𝐴)) | 
| 8 | 1, 7 | syl5 34 | . . . . 5 ⊢ ((Tr 𝐴 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∈ suc 𝐴 → 𝑧 ∈ 𝐴)) | 
| 9 | 8 | expimpd 453 | . . . 4 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ 𝐴)) | 
| 10 | elelsuc 6457 | . . . 4 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) | |
| 11 | 9, 10 | syl6 35 | . . 3 ⊢ (Tr 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) | 
| 12 | 11 | alrimivv 1928 | . 2 ⊢ (Tr 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) | 
| 13 | dftr2 5261 | . 2 ⊢ (Tr suc 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) | |
| 14 | 12, 13 | sylibr 234 | 1 ⊢ (Tr 𝐴 → Tr suc 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∀wal 1538 = wceq 1540 ∈ wcel 2108 Tr wtr 5259 suc csuc 6386 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-sn 4627 df-uni 4908 df-tr 5260 df-suc 6390 | 
| This theorem is referenced by: ordsuci 7828 dfon2lem3 35786 dfon2lem7 35790 dford3lem2 43039 | 
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