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Theorem suctr 6024
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 6007 . . . . . 6 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2 trel 4952 . . . . . . . 8 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
32expdimp 445 . . . . . . 7 ((Tr 𝐴𝑧𝑦) → (𝑦𝐴𝑧𝐴))
4 eleq2 2867 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
54biimpcd 241 . . . . . . . 8 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
65adantl 474 . . . . . . 7 ((Tr 𝐴𝑧𝑦) → (𝑦 = 𝐴𝑧𝐴))
73, 6jaod 886 . . . . . 6 ((Tr 𝐴𝑧𝑦) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧𝐴))
81, 7syl5 34 . . . . 5 ((Tr 𝐴𝑧𝑦) → (𝑦 ∈ suc 𝐴𝑧𝐴))
98expimpd 446 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝐴))
10 elelsuc 6013 . . . 4 (𝑧𝐴𝑧 ∈ suc 𝐴)
119, 10syl6 35 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1211alrimivv 2024 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
13 dftr2 4947 . 2 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1412, 13sylibr 226 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  wo 874  wal 1651   = wceq 1653  wcel 2157  Tr wtr 4945  suc csuc 5943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-un 3774  df-in 3776  df-ss 3783  df-sn 4369  df-uni 4629  df-tr 4946  df-suc 5947
This theorem is referenced by:  dfon2lem3  32202  dfon2lem7  32206  dford3lem2  38379
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