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Theorem suctrALTcf 41626
Description: The sucessor of a transitive class is transitive. suctrALTcf 41626, using conventional notation, was translated from virtual deduction form, suctrALTcfVD 41627, 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 6236 . . . . . . . 8 𝐴 ⊆ suc 𝐴
2 id 22 . . . . . . . . 9 (Tr 𝐴 → Tr 𝐴)
3 id 22 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑧𝑦𝑦 ∈ suc 𝐴))
4 simpl 486 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
53, 4syl 17 . . . . . . . . 9 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
6 id 22 . . . . . . . . 9 (𝑦𝐴𝑦𝐴)
7 trel 5143 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
873impib 1113 . . . . . . . . 9 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
92, 5, 6, 8syl3an 1157 . . . . . . . 8 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧𝐴)
10 ssel2 3910 . . . . . . . 8 ((𝐴 ⊆ suc 𝐴𝑧𝐴) → 𝑧 ∈ suc 𝐴)
111, 9, 10eel0321old 41420 . . . . . . 7 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧 ∈ suc 𝐴)
12113expia 1118 . . . . . 6 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑧 ∈ suc 𝐴))
13 id 22 . . . . . . . . 9 (𝑦 = 𝐴𝑦 = 𝐴)
14 eleq2 2878 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1514biimpac 482 . . . . . . . . 9 ((𝑧𝑦𝑦 = 𝐴) → 𝑧𝐴)
165, 13, 15syl2an 598 . . . . . . . 8 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝐴)
171, 16, 10eel021old 41404 . . . . . . 7 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
1817ex 416 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
19 simpr 488 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
203, 19syl 17 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
21 elsuci 6225 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2220, 21syl 17 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
23 jao 958 . . . . . . 7 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
24233imp 1108 . . . . . 6 (((𝑦𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦 = 𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦𝐴𝑦 = 𝐴)) → 𝑧 ∈ suc 𝐴)
2512, 18, 22, 24eel2122old 41422 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
2625ex 416 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2726alrimivv 1929 . . 3 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
28 dftr2 5138 . . . 4 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2928biimpri 231 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
3027, 29syl 17 . 2 (Tr 𝐴 → Tr suc 𝐴)
3130iin1 41276 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  wal 1536   = wceq 1538  wcel 2111  wss 3881  Tr wtr 5136  suc csuc 6161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-uni 4801  df-tr 5137  df-suc 6165
This theorem is referenced by: (None)
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