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Theorem suctrALTcf 40681
Description: The sucessor of a transitive class is transitive. suctrALTcf 40681, using conventional notation, was translated from virtual deduction form, suctrALTcfVD 40682, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALTcf (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrALTcf
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 6106 . . . . . . . 8 𝐴 ⊆ suc 𝐴
2 id 22 . . . . . . . . 9 (Tr 𝐴 → Tr 𝐴)
3 id 22 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑧𝑦𝑦 ∈ suc 𝐴))
4 simpl 475 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
53, 4syl 17 . . . . . . . . 9 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
6 id 22 . . . . . . . . 9 (𝑦𝐴𝑦𝐴)
7 trel 5037 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
873impib 1096 . . . . . . . . 9 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
92, 5, 6, 8syl3an 1140 . . . . . . . 8 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧𝐴)
10 ssel2 3853 . . . . . . . 8 ((𝐴 ⊆ suc 𝐴𝑧𝐴) → 𝑧 ∈ suc 𝐴)
111, 9, 10eel0321old 40475 . . . . . . 7 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧 ∈ suc 𝐴)
12113expia 1101 . . . . . 6 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑧 ∈ suc 𝐴))
13 id 22 . . . . . . . . 9 (𝑦 = 𝐴𝑦 = 𝐴)
14 eleq2 2854 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1514biimpac 471 . . . . . . . . 9 ((𝑧𝑦𝑦 = 𝐴) → 𝑧𝐴)
165, 13, 15syl2an 586 . . . . . . . 8 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝐴)
171, 16, 10eel021old 40459 . . . . . . 7 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
1817ex 405 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
19 simpr 477 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
203, 19syl 17 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
21 elsuci 6095 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2220, 21syl 17 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
23 jao 943 . . . . . . 7 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
24233imp 1091 . . . . . 6 (((𝑦𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦 = 𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦𝐴𝑦 = 𝐴)) → 𝑧 ∈ suc 𝐴)
2512, 18, 22, 24eel2122old 40477 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
2625ex 405 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2726alrimivv 1887 . . 3 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
28 dftr2 5032 . . . 4 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2928biimpri 220 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
3027, 29syl 17 . 2 (Tr 𝐴 → Tr suc 𝐴)
3130iin1 40331 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wo 833  wal 1505   = wceq 1507  wcel 2050  wss 3829  Tr wtr 5030  suc csuc 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-v 3417  df-un 3834  df-in 3836  df-ss 3843  df-sn 4442  df-uni 4713  df-tr 5031  df-suc 6035
This theorem is referenced by: (None)
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