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Theorem suctrALT 39556
Description: The successor of a transitive class is transitive. The proof of http://us.metamath.org/other/completeusersproof/suctrvd.html is a Virtual Deduction proof verified by automatically transforming it into the Metamath proof of suctrALT 39556 using completeusersproof, which is verified by the Metamath program. The proof of http://us.metamath.org/other/completeusersproof/suctrro.html is a form of the completed proof which preserves the Virtual Deduction proof's step numbers and their ordering. See suctr 6027 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 6021 . . . . . . 7 𝐴 ⊆ suc 𝐴
2 id 22 . . . . . . . 8 (Tr 𝐴 → Tr 𝐴)
3 id 22 . . . . . . . . 9 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑧𝑦𝑦 ∈ suc 𝐴))
43simpld 484 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
5 id 22 . . . . . . . 8 (𝑦𝐴𝑦𝐴)
6 trel 4960 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
763impib 1137 . . . . . . . . 9 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
87idiALT 39182 . . . . . . . 8 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
92, 4, 5, 8syl3an 1192 . . . . . . 7 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧𝐴)
101, 9sseldi 3803 . . . . . 6 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧 ∈ suc 𝐴)
11103expia 1143 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑧 ∈ suc 𝐴))
124adantr 468 . . . . . . . . 9 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝑦)
13 id 22 . . . . . . . . . 10 (𝑦 = 𝐴𝑦 = 𝐴)
1413adantl 469 . . . . . . . . 9 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
1512, 14eleqtrd 2894 . . . . . . . 8 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝐴)
161, 15sseldi 3803 . . . . . . 7 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
1716ex 399 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
1817adantl 469 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
193simprd 485 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
20 elsuci 6010 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2119, 20syl 17 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
2221adantl 469 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑦 = 𝐴))
2311, 18, 22mpjaod 878 . . . 4 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
2423ex 399 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2524alrimivv 2019 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
26 dftr2 4955 . . 3 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2726biimpri 219 . 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 384  wo 865  w3a 1100  wal 1635   = wceq 1637  wcel 2157  Tr wtr 4953  suc csuc 5945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-v 3400  df-un 3781  df-in 3783  df-ss 3790  df-sn 4378  df-uni 4638  df-tr 4954  df-suc 5949
This theorem is referenced by: (None)
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