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Theorem suctr 6472
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 6453 . . . . . 6 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2 trel 5274 . . . . . . . 8 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
32expdimp 452 . . . . . . 7 ((Tr 𝐴𝑧𝑦) → (𝑦𝐴𝑧𝐴))
4 eleq2 2828 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
54biimpcd 249 . . . . . . . 8 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
65adantl 481 . . . . . . 7 ((Tr 𝐴𝑧𝑦) → (𝑦 = 𝐴𝑧𝐴))
73, 6jaod 859 . . . . . 6 ((Tr 𝐴𝑧𝑦) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧𝐴))
81, 7syl5 34 . . . . 5 ((Tr 𝐴𝑧𝑦) → (𝑦 ∈ suc 𝐴𝑧𝐴))
98expimpd 453 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝐴))
10 elelsuc 6459 . . . 4 (𝑧𝐴𝑧 ∈ suc 𝐴)
119, 10syl6 35 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1211alrimivv 1926 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
13 dftr2 5267 . 2 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1412, 13sylibr 234 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  wal 1535   = wceq 1537  wcel 2106  Tr wtr 5265  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-uni 4913  df-tr 5266  df-suc 6392
This theorem is referenced by:  ordsuci  7828  dfon2lem3  35767  dfon2lem7  35771  dford3lem2  43016
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