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Theorem suctr 5725
 Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.) (Proof shortened by JJ, 24-Sep-2021.)
Assertion
Ref Expression
suctr (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctr
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsuci 5708 . . . . . 6 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2 trel 4687 . . . . . . . 8 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
32expdimp 452 . . . . . . 7 ((Tr 𝐴𝑧𝑦) → (𝑦𝐴𝑧𝐴))
4 eleq2 2677 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
54biimpcd 238 . . . . . . . 8 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
65adantl 481 . . . . . . 7 ((Tr 𝐴𝑧𝑦) → (𝑦 = 𝐴𝑧𝐴))
73, 6jaod 394 . . . . . 6 ((Tr 𝐴𝑧𝑦) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧𝐴))
81, 7syl5 33 . . . . 5 ((Tr 𝐴𝑧𝑦) → (𝑦 ∈ suc 𝐴𝑧𝐴))
98expimpd 627 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝐴))
10 elelsuc 5714 . . . 4 (𝑧𝐴𝑧 ∈ suc 𝐴)
119, 10syl6 34 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1211alrimivv 1843 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
13 dftr2 4682 . 2 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1412, 13sylibr 223 1 (Tr 𝐴 → Tr suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Tr wtr 4680  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-uni 4373  df-tr 4681  df-suc 5646 This theorem is referenced by:  dfon2lem3  30934  dfon2lem7  30938  dford3lem2  36612
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