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Theorem suctrALT 42399
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 42399 using completeusersproof, which is verified by the Metamath program. The proof of https://us.metamath.org/other/completeusersproof/suctrro.html 42399 is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. See suctr 6346 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 6340 . . . . . . 7 𝐴 ⊆ suc 𝐴
2 id 22 . . . . . . . 8 (Tr 𝐴 → Tr 𝐴)
3 id 22 . . . . . . . . 9 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑧𝑦𝑦 ∈ suc 𝐴))
43simpld 494 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
5 id 22 . . . . . . . 8 (𝑦𝐴𝑦𝐴)
6 trel 5202 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
763impib 1114 . . . . . . . . 9 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
87idiALT 42050 . . . . . . . 8 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
92, 4, 5, 8syl3an 1158 . . . . . . 7 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧𝐴)
101, 9sselid 3923 . . . . . 6 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧 ∈ suc 𝐴)
11103expia 1119 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑧 ∈ suc 𝐴))
124adantr 480 . . . . . . . . 9 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝑦)
13 id 22 . . . . . . . . . 10 (𝑦 = 𝐴𝑦 = 𝐴)
1413adantl 481 . . . . . . . . 9 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
1512, 14eleqtrd 2842 . . . . . . . 8 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝐴)
161, 15sselid 3923 . . . . . . 7 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
1716ex 412 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
1817adantl 481 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
193simprd 495 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
20 elsuci 6329 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2119, 20syl 17 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
2221adantl 481 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑦 = 𝐴))
2311, 18, 22mpjaod 856 . . . 4 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
2423ex 412 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2524alrimivv 1934 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
26 dftr2 5197 . . 3 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2726biimpri 227 . 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 395  wo 843  w3a 1085  wal 1539   = wceq 1541  wcel 2109  Tr wtr 5195  suc csuc 6265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-v 3432  df-un 3896  df-in 3898  df-ss 3908  df-sn 4567  df-uni 4845  df-tr 5196  df-suc 6269
This theorem is referenced by: (None)
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