MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suctr Structured version   Visualization version   GIF version

Theorem suctr 5777
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 5760 . . . . . 6 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2 trel 4729 . . . . . . . 8 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
32expdimp 453 . . . . . . 7 ((Tr 𝐴𝑧𝑦) → (𝑦𝐴𝑧𝐴))
4 eleq2 2687 . . . . . . . . 9 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
54biimpcd 239 . . . . . . . 8 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
65adantl 482 . . . . . . 7 ((Tr 𝐴𝑧𝑦) → (𝑦 = 𝐴𝑧𝐴))
73, 6jaod 395 . . . . . 6 ((Tr 𝐴𝑧𝑦) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧𝐴))
81, 7syl5 34 . . . . 5 ((Tr 𝐴𝑧𝑦) → (𝑦 ∈ suc 𝐴𝑧𝐴))
98expimpd 628 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝐴))
10 elelsuc 5766 . . . 4 (𝑧𝐴𝑧 ∈ suc 𝐴)
119, 10syl6 35 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1211alrimivv 1853 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
13 dftr2 4724 . 2 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1412, 13sylibr 224 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  wal 1478   = wceq 1480  wcel 1987  Tr wtr 4722  suc csuc 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-un 3565  df-in 3567  df-ss 3574  df-sn 4156  df-uni 4410  df-tr 4723  df-suc 5698
This theorem is referenced by:  dfon2lem3  31444  dfon2lem7  31448  dford3lem2  37113
  Copyright terms: Public domain W3C validator