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Theorem suctrALT2VD 42785
Description: Virtual deduction proof of suctrALT2 42786. (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 5211 . . 3 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2 sssucid 6381 . . . . . . . 8 𝐴 ⊆ suc 𝐴
3 idn1 42523 . . . . . . . . 9 (   Tr 𝐴   ▶   Tr 𝐴   )
4 idn2 42562 . . . . . . . . . 10 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑧𝑦𝑦 ∈ suc 𝐴)   )
5 simpl 483 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
64, 5e2 42580 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧𝑦   )
7 idn3 42564 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑦𝐴   )
8 trel 5218 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
98expd 416 . . . . . . . . 9 (Tr 𝐴 → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
103, 6, 7, 9e123 42711 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑧𝐴   )
11 ssel 3925 . . . . . . . 8 (𝐴 ⊆ suc 𝐴 → (𝑧𝐴𝑧 ∈ suc 𝐴))
122, 10, 11e03 42689 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑧 ∈ suc 𝐴   )
1312in3 42558 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑧 ∈ suc 𝐴)   )
14 idn3 42564 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
15 eleq2 2825 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1615biimpcd 248 . . . . . . . . 9 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
176, 14, 16e23 42704 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧𝐴   )
182, 17, 11e03 42689 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧 ∈ suc 𝐴   )
1918in3 42558 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴𝑧 ∈ suc 𝐴)   )
20 simpr 485 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
214, 20e2 42580 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
22 elsuci 6368 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2321, 22e2 42580 . . . . . 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 42585 . . . . 5 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ suc 𝐴   )
2625in2 42554 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
2726gen12 42567 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
28 biimpr 219 . . 3 ((Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴))
291, 27, 28e01 42640 . 2 (   Tr 𝐴   ▶   Tr suc 𝐴   )
3029in1 42520 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844  wal 1538   = wceq 1540  wcel 2105  wss 3898  Tr wtr 5209  suc csuc 6304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-un 3903  df-in 3905  df-ss 3915  df-sn 4574  df-uni 4853  df-tr 5210  df-suc 6308  df-vd1 42519  df-vd2 42527  df-vd3 42539
This theorem is referenced by: (None)
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