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Theorem suctrALTcfVD 42496
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 42142) 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 42495 is suctrALTcfVD 42496 without virtual deductions and was derived automatically from suctrALTcfVD 42496. 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 6340 . . . . . . . 8 𝐴 ⊆ suc 𝐴
2 idn1 42147 . . . . . . . . 9 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn1 42147 . . . . . . . . . 10 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑧𝑦𝑦 ∈ suc 𝐴)   )
4 simpl 482 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
53, 4el1 42201 . . . . . . . . 9 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧𝑦   )
6 idn1 42147 . . . . . . . . 9 (   𝑦𝐴   ▶   𝑦𝐴   )
7 trel 5202 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
873impib 1114 . . . . . . . . 9 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
92, 5, 6, 8el123 42337 . . . . . . . 8 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   )   ▶   𝑧𝐴   )
10 ssel2 3920 . . . . . . . 8 ((𝐴 ⊆ suc 𝐴𝑧𝐴) → 𝑧 ∈ suc 𝐴)
111, 9, 10el0321old 42290 . . . . . . 7 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
1211int3 42185 . . . . . 6 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   )   ▶   (𝑦𝐴𝑧 ∈ suc 𝐴)   )
13 idn1 42147 . . . . . . . . 9 (   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
14 eleq2 2828 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1514biimpac 478 . . . . . . . . 9 ((𝑧𝑦𝑦 = 𝐴) → 𝑧𝐴)
165, 13, 15el12 42299 . . . . . . . 8 (   (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧𝐴   )
171, 16, 10el021old 42274 . . . . . . 7 (   (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
1817int2 42179 . . . . . 6 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴𝑧 ∈ suc 𝐴)   )
19 simpr 484 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
203, 19el1 42201 . . . . . . 7 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
21 elsuci 6329 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2220, 21el1 42201 . . . . . 6 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑦 = 𝐴)   )
23 jao 957 . . . . . . 7 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
24233imp 1109 . . . . . 6 (((𝑦𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦 = 𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦𝐴𝑦 = 𝐴)) → 𝑧 ∈ suc 𝐴)
2512, 18, 22, 24el2122old 42292 . . . . 5 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   )   ▶   𝑧 ∈ suc 𝐴   )
2625int2 42179 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
2726gen12 42191 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
28 dftr2 5197 . . . 4 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2928biimpri 227 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
3027, 29el1 42201 . 2 (   Tr 𝐴   ▶   Tr suc 𝐴   )
3130in1 42144 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 843  wal 1539   = wceq 1541  wcel 2109  wss 3891  Tr wtr 5195  suc csuc 6265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-v 3432  df-un 3896  df-in 3898  df-ss 3908  df-sn 4567  df-uni 4845  df-tr 5196  df-suc 6269  df-vd1 42143  df-vhc2 42154  df-vhc3 42162
This theorem is referenced by: (None)
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