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Theorem suctrALTcfVD 39473
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 39102) 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 39472 is suctrALTcfVD 39473 without virtual deductions and was derived automatically from suctrALTcfVD 39473. 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 5840 . . . . . . . 8 𝐴 ⊆ suc 𝐴
2 idn1 39107 . . . . . . . . 9 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn1 39107 . . . . . . . . . 10 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑧𝑦𝑦 ∈ suc 𝐴)   )
4 simpl 472 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
53, 4el1 39170 . . . . . . . . 9 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧𝑦   )
6 idn1 39107 . . . . . . . . 9 (   𝑦𝐴   ▶   𝑦𝐴   )
7 trel 4792 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
873impib 1281 . . . . . . . . 9 ((Tr 𝐴𝑧𝑦𝑦𝐴) → 𝑧𝐴)
92, 5, 6, 8el123 39308 . . . . . . . 8 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   )   ▶   𝑧𝐴   )
10 ssel2 3631 . . . . . . . 8 ((𝐴 ⊆ suc 𝐴𝑧𝐴) → 𝑧 ∈ suc 𝐴)
111, 9, 10el0321old 39259 . . . . . . 7 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
1211int3 39154 . . . . . 6 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   )   ▶   (𝑦𝐴𝑧 ∈ suc 𝐴)   )
13 idn1 39107 . . . . . . . . 9 (   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
14 eleq2 2719 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1514biimpac 502 . . . . . . . . 9 ((𝑧𝑦𝑦 = 𝐴) → 𝑧𝐴)
165, 13, 15el12 39270 . . . . . . . 8 (   (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧𝐴   )
171, 16, 10el021old 39243 . . . . . . 7 (   (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   )   ▶   𝑧 ∈ suc 𝐴   )
1817int2 39148 . . . . . 6 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴𝑧 ∈ suc 𝐴)   )
19 simpr 476 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
203, 19el1 39170 . . . . . . 7 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
21 elsuci 5829 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2220, 21el1 39170 . . . . . 6 (   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑦 = 𝐴)   )
23 jao 533 . . . . . . 7 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
24233imp 1275 . . . . . 6 (((𝑦𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦 = 𝐴𝑧 ∈ suc 𝐴) ∧ (𝑦𝐴𝑦 = 𝐴)) → 𝑧 ∈ suc 𝐴)
2512, 18, 22, 24el2122old 39261 . . . . 5 (   (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   )   ▶   𝑧 ∈ suc 𝐴   )
2625int2 39148 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
2726gen12 39160 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
28 dftr2 4787 . . . 4 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2928biimpri 218 . . 3 (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴)
3027, 29el1 39170 . 2 (   Tr 𝐴   ▶   Tr suc 𝐴   )
3130in1 39104 1 (Tr 𝐴 → Tr suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383  ∀wal 1521   = wceq 1523   ∈ wcel 2030   ⊆ wss 3607  Tr wtr 4785  suc csuc 5763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-sn 4211  df-uni 4469  df-tr 4786  df-suc 5767  df-vd1 39103  df-vhc2 39114  df-vhc3 39122 This theorem is referenced by: (None)
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