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Theorem suctrALT2VD 39563
Description: Virtual deduction proof of suctrALT2 39564. (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 4946 . . 3 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2 sssucid 6013 . . . . . . . 8 𝐴 ⊆ suc 𝐴
3 idn1 39286 . . . . . . . . 9 (   Tr 𝐴   ▶   Tr 𝐴   )
4 idn2 39334 . . . . . . . . . 10 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑧𝑦𝑦 ∈ suc 𝐴)   )
5 simpl 470 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
64, 5e2 39352 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧𝑦   )
7 idn3 39336 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑦𝐴   )
8 trel 4951 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
98expd 402 . . . . . . . . 9 (Tr 𝐴 → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
103, 6, 7, 9e123 39484 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑧𝐴   )
11 ssel 3790 . . . . . . . 8 (𝐴 ⊆ suc 𝐴 → (𝑧𝐴𝑧 ∈ suc 𝐴))
122, 10, 11e03 39462 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑧 ∈ suc 𝐴   )
1312in3 39330 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑧 ∈ suc 𝐴)   )
14 idn3 39336 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
15 eleq2 2872 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1615biimpcd 240 . . . . . . . . 9 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
176, 14, 16e23 39477 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧𝐴   )
182, 17, 11e03 39462 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧 ∈ suc 𝐴   )
1918in3 39330 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴𝑧 ∈ suc 𝐴)   )
20 simpr 473 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
214, 20e2 39352 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
22 elsuci 6002 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2321, 22e2 39352 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑦 = 𝐴)   )
24 jao 974 . . . . . 6 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
2513, 19, 23, 24e222 39357 . . . . 5 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ suc 𝐴   )
2625in2 39326 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
2726gen12 39339 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
28 biimpr 211 . . 3 ((Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴))
291, 27, 28e01 39412 . 2 (   Tr 𝐴   ▶   Tr suc 𝐴   )
3029in1 39283 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 865  wal 1635   = wceq 1637  wcel 2156  wss 3767  Tr wtr 4944  suc csuc 5936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-v 3391  df-un 3772  df-in 3774  df-ss 3781  df-sn 4369  df-uni 4629  df-tr 4945  df-suc 5940  df-vd1 39282  df-vd2 39290  df-vd3 39302
This theorem is referenced by: (None)
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