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Theorem suctrALT3 39677
Description: The successor of a transitive class is transitive. suctrALT3 39677 is the completed proof in conventional notation of the Virtual Deduction proof http://us.metamath.org/other/completeusersproof/suctralt3vd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 39305 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction (e.g., the sub-theorem whose assertion is step 19 used jaoded 39302). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem (e.g., the sub-theorem whose assertion is step 24 used dftr2 4906) . (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALT3 (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrALT3
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 5963 . . . . . . . . 9 𝐴 ⊆ suc 𝐴
2 id 22 . . . . . . . . . 10 (Tr 𝐴 → Tr 𝐴)
3 id 22 . . . . . . . . . . 11 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑧𝑦𝑦 ∈ suc 𝐴))
43simpld 477 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
5 id 22 . . . . . . . . . 10 (𝑦𝐴𝑦𝐴)
62, 4, 5trelded 39301 . . . . . . . . 9 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧𝐴)
71, 6sseldi 3742 . . . . . . . 8 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧 ∈ suc 𝐴)
873expia 1115 . . . . . . 7 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑧 ∈ suc 𝐴))
9 id 22 . . . . . . . . . 10 (𝑦 = 𝐴𝑦 = 𝐴)
10 eleq2 2828 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1110biimpac 504 . . . . . . . . . 10 ((𝑧𝑦𝑦 = 𝐴) → 𝑧𝐴)
124, 9, 11syl2an 495 . . . . . . . . 9 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝐴)
131, 12sseldi 3742 . . . . . . . 8 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
1413ex 449 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
153simprd 482 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
16 elsuci 5952 . . . . . . . 8 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
1715, 16syl 17 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
188, 14, 17jaoded 39302 . . . . . 6 (((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
1918un2122 39537 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
2019ex 449 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2120alrimivv 2005 . . 3 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
22 dftr2 4906 . . . 4 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2322biimpri 218 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
2421, 23syl 17 . 2 (Tr 𝐴 → Tr suc 𝐴)
2524idiALT 39203 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1072  wal 1630   = wceq 1632  wcel 2139  Tr wtr 4904  suc csuc 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-in 3722  df-ss 3729  df-sn 4322  df-uni 4589  df-tr 4905  df-suc 5890
This theorem is referenced by: (None)
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