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Theorem suctrALT2VD 41629
 Description: Virtual deduction proof of suctrALT2 41630. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALT2VD (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrALT2VD
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 5141 . . 3 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2 sssucid 6241 . . . . . . . 8 𝐴 ⊆ suc 𝐴
3 idn1 41367 . . . . . . . . 9 (   Tr 𝐴   ▶   Tr 𝐴   )
4 idn2 41406 . . . . . . . . . 10 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑧𝑦𝑦 ∈ suc 𝐴)   )
5 simpl 486 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
64, 5e2 41424 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧𝑦   )
7 idn3 41408 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑦𝐴   )
8 trel 5146 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
98expd 419 . . . . . . . . 9 (Tr 𝐴 → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
103, 6, 7, 9e123 41555 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑧𝐴   )
11 ssel 3909 . . . . . . . 8 (𝐴 ⊆ suc 𝐴 → (𝑧𝐴𝑧 ∈ suc 𝐴))
122, 10, 11e03 41533 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑧 ∈ suc 𝐴   )
1312in3 41402 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑧 ∈ suc 𝐴)   )
14 idn3 41408 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
15 eleq2 2878 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1615biimpcd 252 . . . . . . . . 9 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
176, 14, 16e23 41548 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧𝐴   )
182, 17, 11e03 41533 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧 ∈ suc 𝐴   )
1918in3 41402 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴𝑧 ∈ suc 𝐴)   )
20 simpr 488 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
214, 20e2 41424 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
22 elsuci 6230 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2321, 22e2 41424 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑦 = 𝐴)   )
24 jao 958 . . . . . 6 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
2513, 19, 23, 24e222 41429 . . . . 5 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ suc 𝐴   )
2625in2 41398 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
2726gen12 41411 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
28 biimpr 223 . . 3 ((Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴))
291, 27, 28e01 41484 . 2 (   Tr 𝐴   ▶   Tr suc 𝐴   )
3029in1 41364 1 (Tr 𝐴 → Tr suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538   ∈ wcel 2111   ⊆ wss 3882  Tr wtr 5139  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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3887  df-in 3889  df-ss 3899  df-sn 4528  df-uni 4804  df-tr 5140  df-suc 6170  df-vd1 41363  df-vd2 41371  df-vd3 41383 This theorem is referenced by: (None)
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