Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  suctrALT Structured version   Visualization version   GIF version

Theorem suctrALT 41343
Description: The successor of a transitive class is transitive. The proof of https://us.metamath.org/other/completeusersproof/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctrALT 41343 using completeusersproof, which is verified by the Metamath program. The proof of https://us.metamath.org/other/completeusersproof/suctrro.html 41343 is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. See suctr 6247 for the original proof. (Contributed by Alan Sare, 11-Apr-2009.) (Revised by Alan Sare, 12-Jun-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALT (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrALT
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 6241 . . . . . . 7 𝐴 ⊆ suc 𝐴
2 id 22 . . . . . . . 8 (Tr 𝐴 → Tr 𝐴)
3 id 22 . . . . . . . . 9 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑧𝑦𝑦 ∈ suc 𝐴))
43simpld 498 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
5 id 22 . . . . . . . 8 (𝑦𝐴𝑦𝐴)
6 trel 5152 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
763impib 1113 . . . . . . . . 9 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
87idiALT 40994 . . . . . . . 8 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
92, 4, 5, 8syl3an 1157 . . . . . . 7 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧𝐴)
101, 9sseldi 3941 . . . . . 6 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧 ∈ suc 𝐴)
11103expia 1118 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑧 ∈ suc 𝐴))
124adantr 484 . . . . . . . . 9 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝑦)
13 id 22 . . . . . . . . . 10 (𝑦 = 𝐴𝑦 = 𝐴)
1413adantl 485 . . . . . . . . 9 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
1512, 14eleqtrd 2914 . . . . . . . 8 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝐴)
161, 15sseldi 3941 . . . . . . 7 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
1716ex 416 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
1817adantl 485 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
193simprd 499 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
20 elsuci 6230 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2119, 20syl 17 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
2221adantl 485 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑦 = 𝐴))
2311, 18, 22mpjaod 857 . . . 4 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
2423ex 416 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2524alrimivv 1930 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
26 dftr2 5147 . . 3 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2726biimpri 231 . 2 (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
2825, 27syl 17 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084  wal 1536   = wceq 1538  wcel 2115  Tr wtr 5145  suc csuc 6166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473  df-un 3915  df-in 3917  df-ss 3927  df-sn 4541  df-uni 4812  df-tr 5146  df-suc 6170
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator