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Theorem suctrALT3 38079
Description: The successor of a transitive class is transitive. suctrALT3 38079 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 37703 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 37700). 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 4580) . (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 5607 . . . . . . . . 9 𝐴 ⊆ suc 𝐴
2 id 22 . . . . . . . . . 10 (Tr 𝐴 → Tr 𝐴)
3 id 22 . . . . . . . . . . 11 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑧𝑦𝑦 ∈ suc 𝐴))
43simpld 473 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
5 id 22 . . . . . . . . . 10 (𝑦𝐴𝑦𝐴)
62, 4, 5trelded 37699 . . . . . . . . 9 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧𝐴)
71, 6sseldi 3470 . . . . . . . 8 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧 ∈ suc 𝐴)
873expia 1258 . . . . . . 7 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑧 ∈ suc 𝐴))
9 id 22 . . . . . . . . . 10 (𝑦 = 𝐴𝑦 = 𝐴)
10 eleq2 2581 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1110biimpac 501 . . . . . . . . . 10 ((𝑧𝑦𝑦 = 𝐴) → 𝑧𝐴)
124, 9, 11syl2an 492 . . . . . . . . 9 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝐴)
131, 12sseldi 3470 . . . . . . . 8 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
1413ex 448 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
153simprd 477 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
16 elsuci 5596 . . . . . . . 8 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
1715, 16syl 17 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
188, 14, 17jaoded 37700 . . . . . 6 (((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
1918un2122 37935 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
2019ex 448 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2120alrimivv 1809 . . 3 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
22 dftr2 4580 . . . 4 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2322biimpri 216 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
2421, 23syl 17 . 2 (Tr 𝐴 → Tr suc 𝐴)
2524idiALT 37601 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 381  wa 382  w3a 1030  wal 1472   = wceq 1474  wcel 1938  Tr wtr 4578  suc csuc 5532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-v 3079  df-un 3449  df-in 3451  df-ss 3458  df-sn 4029  df-uni 4271  df-tr 4579  df-suc 5536
This theorem is referenced by: (None)
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