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Theorem suctrALT3 39971
Description: The successor of a transitive class is transitive. suctrALT3 39971 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 39606 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 39603). 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 4977) . (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 6040 . . . . . . . . 9 𝐴 ⊆ suc 𝐴
2 id 22 . . . . . . . . . 10 (Tr 𝐴 → Tr 𝐴)
3 id 22 . . . . . . . . . . 11 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑧𝑦𝑦 ∈ suc 𝐴))
43simpld 490 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
5 id 22 . . . . . . . . . 10 (𝑦𝐴𝑦𝐴)
62, 4, 5trelded 39602 . . . . . . . . 9 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧𝐴)
71, 6sseldi 3825 . . . . . . . 8 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧 ∈ suc 𝐴)
873expia 1154 . . . . . . 7 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑧 ∈ suc 𝐴))
9 id 22 . . . . . . . . . 10 (𝑦 = 𝐴𝑦 = 𝐴)
10 eleq2 2895 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1110biimpac 472 . . . . . . . . . 10 ((𝑧𝑦𝑦 = 𝐴) → 𝑧𝐴)
124, 9, 11syl2an 589 . . . . . . . . 9 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝐴)
131, 12sseldi 3825 . . . . . . . 8 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
1413ex 403 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
153simprd 491 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
16 elsuci 6029 . . . . . . . 8 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
1715, 16syl 17 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
188, 14, 17jaoded 39603 . . . . . 6 (((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
1918un2122 39837 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
2019ex 403 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2120alrimivv 2027 . . 3 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
22 dftr2 4977 . . . 4 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2322biimpri 220 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
2421, 23syl 17 . 2 (Tr 𝐴 → Tr suc 𝐴)
2524idiALT 39514 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wo 878  w3a 1111  wal 1654   = wceq 1656  wcel 2164  Tr wtr 4975  suc csuc 5965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-un 3803  df-in 3805  df-ss 3812  df-sn 4398  df-uni 4659  df-tr 4976  df-suc 5969
This theorem is referenced by: (None)
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