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Theorem suctrALTcfVD 41414
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 41060) using conjunction-form virtual hypothesis collections. The conjunction-form version of completeusersproof.cmd. It allows the User to avoid superflous virtual hypotheses. This proof was completed automatically by a tools program which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. suctrALTcf 41413 is suctrALTcfVD 41414 without virtual deductions and was derived automatically from suctrALTcfVD 41414. The version of completeusersproof.cmd used is capable of only generating conjunction-form unification theorems, not unification deductions. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: ⊢ (   Tr 𝐴   ▶   Tr 𝐴   ) 2:: ⊢ (   ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ) 3:2: ⊢ (   ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ 𝑦   ) 4:: ⊢ (   ................................... ....... 𝑦 ∈ 𝐴   ▶   𝑦 ∈ 𝐴   ) 5:1,3,4: ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) , 𝑦 ∈ 𝐴   )   ▶   𝑧 ∈ 𝐴   ) 6:: ⊢ 𝐴 ⊆ suc 𝐴 7:5,6: ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) , 𝑦 ∈ 𝐴   )   ▶   𝑧 ∈ suc 𝐴   ) 8:7: ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)    )   ▶   (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴)   ) 9:: ⊢ (   ................................... ...... 𝑦 = 𝐴   ▶   𝑦 = 𝐴   ) 10:3,9: ⊢ (   ........ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴), 𝑦 = 𝐴   )   ▶   𝑧 ∈ 𝐴   ) 11:10,6: ⊢ (   ........ (   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴), 𝑦 = 𝐴   )   ▶   𝑧 ∈ suc 𝐴   ) 12:11: ⊢ (   .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴)   ) 13:2: ⊢ (   .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   ) 14:13: ⊢ (   .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)   ▶   (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴)   ) 15:8,12,14: ⊢ (   (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)    )   ▶   𝑧 ∈ suc 𝐴   ) 16:15: ⊢ (   Tr 𝐴   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   ) 17:16: ⊢ (   Tr 𝐴   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   ) 18:17: ⊢ (   Tr 𝐴   ▶   Tr suc 𝐴   ) qed:18: ⊢ (Tr 𝐴 → Tr suc 𝐴)
Assertion
Ref Expression
suctrALTcfVD (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrALTcfVD
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 6241 . . . . . . . 8 𝐴 ⊆ suc 𝐴
2 idn1 41065 . . . . . . . . 9 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn1 41065 . . . . . . . . . 10 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑧𝑦𝑦 ∈ suc 𝐴)   )
4 simpl 486 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
53, 4el1 41119 . . . . . . . . 9 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧𝑦   )
6 idn1 41065 . . . . . . . . 9 (   𝑦𝐴   ▶   𝑦𝐴   )
7 trel 5152 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
873impib 1113 . . . . . . . . 9 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
92, 5, 6, 8el123 41255 . . . . . . . 8 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   )   ▶   𝑧𝐴   )
10 ssel2 3938 . . . . . . . 8 ((𝐴 ⊆ suc 𝐴𝑧𝐴) → 𝑧 ∈ suc 𝐴)
111, 9, 10el0321old 41208 . . . . . . 7 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
1211int3 41103 . . . . . 6 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   )   ▶   (𝑦𝐴𝑧 ∈ suc 𝐴)   )
13 idn1 41065 . . . . . . . . 9 (   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
14 eleq2 2900 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1514biimpac 482 . . . . . . . . 9 ((𝑧𝑦𝑦 = 𝐴) → 𝑧𝐴)
165, 13, 15el12 41217 . . . . . . . 8 (   (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧𝐴   )
171, 16, 10el021old 41192 . . . . . . 7 (   (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
1817int2 41097 . . . . . 6 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴𝑧 ∈ suc 𝐴)   )
19 simpr 488 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
203, 19el1 41119 . . . . . . 7 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
21 elsuci 6230 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2220, 21el1 41119 . . . . . 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, 24el2122old 41210 . . . . 5 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   )   ▶   𝑧 ∈ suc 𝐴   )
2625int2 41097 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
2726gen12 41109 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
28 dftr2 5147 . . . 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, 29el1 41119 . 2 (   Tr 𝐴   ▶   Tr suc 𝐴   )
3130in1 41062 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 2115   ⊆ wss 3910  Tr wtr 5145  suc csuc 6166 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473  df-un 3915  df-in 3917  df-ss 3927  df-sn 4541  df-uni 4812  df-tr 5146  df-suc 6170  df-vd1 41061  df-vhc2 41072  df-vhc3 41080 This theorem is referenced by: (None)
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