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Theorem suctrALTcfVD 44921
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 44567) 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 44920 is suctrALTcfVD 44921 without virtual deductions and was derived automatically from suctrALTcfVD 44921. 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:: (   ......... (𝑧𝑦𝑦 suc 𝐴)   ▶   (𝑧𝑦𝑦 ∈ suc 𝐴)   )
3:2: (   ......... (𝑧𝑦𝑦 suc 𝐴)   ▶   𝑧𝑦   )
4:: (   ................................... ....... 𝑦𝐴   ▶   𝑦𝐴   )
5:1,3,4: (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴) , 𝑦𝐴   )   ▶   𝑧𝐴   )
6:: 𝐴 ⊆ suc 𝐴
7:5,6: (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴) , 𝑦𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
8:7: (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)    )   ▶   (𝑦𝐴𝑧 ∈ suc 𝐴)   )
9:: (   ................................... ...... 𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
10:3,9: (   ........ (   (𝑧𝑦𝑦 suc 𝐴), 𝑦 = 𝐴   )   ▶   𝑧𝐴   )
11:10,6: (   ........ (   (𝑧𝑦𝑦 suc 𝐴), 𝑦 = 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
12:11: (   .......... (𝑧𝑦𝑦 suc 𝐴)   ▶   (𝑦 = 𝐴𝑧 ∈ suc 𝐴)   )
13:2: (   .......... (𝑧𝑦𝑦 suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
14:13: (   .......... (𝑧𝑦𝑦 suc 𝐴)   ▶   (𝑦𝐴𝑦 = 𝐴)   )
15:8,12,14: (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)    )   ▶   𝑧 ∈ suc 𝐴   )
16:15: (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
17:16: (   Tr 𝐴   ▶   𝑧𝑦((𝑧 𝑦𝑦 ∈ 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 6466 . . . . . . . 8 𝐴 ⊆ suc 𝐴
2 idn1 44572 . . . . . . . . 9 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn1 44572 . . . . . . . . . 10 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑧𝑦𝑦 ∈ suc 𝐴)   )
4 simpl 482 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
53, 4el1 44626 . . . . . . . . 9 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧𝑦   )
6 idn1 44572 . . . . . . . . 9 (   𝑦𝐴   ▶   𝑦𝐴   )
7 trel 5274 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
873impib 1115 . . . . . . . . 9 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
92, 5, 6, 8el123 44762 . . . . . . . 8 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   )   ▶   𝑧𝐴   )
10 ssel2 3990 . . . . . . . 8 ((𝐴 ⊆ suc 𝐴𝑧𝐴) → 𝑧 ∈ suc 𝐴)
111, 9, 10el0321old 44715 . . . . . . 7 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
1211int3 44610 . . . . . 6 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   )   ▶   (𝑦𝐴𝑧 ∈ suc 𝐴)   )
13 idn1 44572 . . . . . . . . 9 (   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
14 eleq2 2828 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1514biimpac 478 . . . . . . . . 9 ((𝑧𝑦𝑦 = 𝐴) → 𝑧𝐴)
165, 13, 15el12 44724 . . . . . . . 8 (   (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧𝐴   )
171, 16, 10el021old 44699 . . . . . . 7 (   (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
1817int2 44604 . . . . . 6 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴𝑧 ∈ suc 𝐴)   )
19 simpr 484 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
203, 19el1 44626 . . . . . . 7 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
21 elsuci 6453 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2220, 21el1 44626 . . . . . 6 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑦 = 𝐴)   )
23 jao 962 . . . . . . 7 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
24233imp 1110 . . . . . 6 (((𝑦𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦 = 𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦𝐴𝑦 = 𝐴)) → 𝑧 ∈ suc 𝐴)
2512, 18, 22, 24el2122old 44717 . . . . 5 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   )   ▶   𝑧 ∈ suc 𝐴   )
2625int2 44604 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
2726gen12 44616 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
28 dftr2 5267 . . . 4 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2928biimpri 228 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
3027, 29el1 44626 . 2 (   Tr 𝐴   ▶   Tr suc 𝐴   )
3130in1 44569 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  wal 1535   = wceq 1537  wcel 2106  wss 3963  Tr wtr 5265  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-uni 4913  df-tr 5266  df-suc 6392  df-vd1 44568  df-vhc2 44579  df-vhc3 44587
This theorem is referenced by: (None)
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