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Theorem suctrALTcf 41133
Description: The sucessor of a transitive class is transitive. suctrALTcf 41133, using conventional notation, was translated from virtual deduction form, suctrALTcfVD 41134, 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 6261 . . . . . . . 8 𝐴 ⊆ suc 𝐴
2 id 22 . . . . . . . . 9 (Tr 𝐴 → Tr 𝐴)
3 id 22 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑧𝑦𝑦 ∈ suc 𝐴))
4 simpl 483 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
53, 4syl 17 . . . . . . . . 9 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
6 id 22 . . . . . . . . 9 (𝑦𝐴𝑦𝐴)
7 trel 5170 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
873impib 1108 . . . . . . . . 9 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
92, 5, 6, 8syl3an 1152 . . . . . . . 8 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧𝐴)
10 ssel2 3959 . . . . . . . 8 ((𝐴 ⊆ suc 𝐴𝑧𝐴) → 𝑧 ∈ suc 𝐴)
111, 9, 10eel0321old 40927 . . . . . . 7 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦𝐴) → 𝑧 ∈ suc 𝐴)
12113expia 1113 . . . . . 6 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → (𝑦𝐴𝑧 ∈ suc 𝐴))
13 id 22 . . . . . . . . 9 (𝑦 = 𝐴𝑦 = 𝐴)
14 eleq2 2898 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1514biimpac 479 . . . . . . . . 9 ((𝑧𝑦𝑦 = 𝐴) → 𝑧𝐴)
165, 13, 15syl2an 595 . . . . . . . 8 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧𝐴)
171, 16, 10eel021old 40911 . . . . . . 7 (((𝑧𝑦𝑦 ∈ suc 𝐴) ∧ 𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)
1817ex 413 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
19 simpr 485 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
203, 19syl 17 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
21 elsuci 6250 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2220, 21syl 17 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
23 jao 954 . . . . . . 7 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
24233imp 1103 . . . . . 6 (((𝑦𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦 = 𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦𝐴𝑦 = 𝐴)) → 𝑧 ∈ suc 𝐴)
2512, 18, 22, 24eel2122old 40929 . . . . 5 ((Tr 𝐴 ∧ (𝑧𝑦𝑦 ∈ suc 𝐴)) → 𝑧 ∈ suc 𝐴)
2625ex 413 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2726alrimivv 1920 . . 3 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
28 dftr2 5165 . . . 4 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2928biimpri 229 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
3027, 29syl 17 . 2 (Tr 𝐴 → Tr suc 𝐴)
3130iin1 40783 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841  wal 1526   = wceq 1528  wcel 2105  wss 3933  Tr wtr 5163  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-un 3938  df-in 3940  df-ss 3949  df-sn 4558  df-uni 4831  df-tr 5164  df-suc 6190
This theorem is referenced by: (None)
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