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Theorem suctrALT2VD 39385
Description: Virtual deduction proof of suctrALT2 39386. (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 4787 . . 3 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2 sssucid 5840 . . . . . . . 8 𝐴 ⊆ suc 𝐴
3 idn1 39107 . . . . . . . . 9 (   Tr 𝐴   ▶   Tr 𝐴   )
4 idn2 39155 . . . . . . . . . 10 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑧𝑦𝑦 ∈ suc 𝐴)   )
5 simpl 472 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
64, 5e2 39173 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧𝑦   )
7 idn3 39157 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑦𝐴   )
8 trel 4792 . . . . . . . . . 10 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
98expd 451 . . . . . . . . 9 (Tr 𝐴 → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
103, 6, 7, 9e123 39306 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑧𝐴   )
11 ssel 3630 . . . . . . . 8 (𝐴 ⊆ suc 𝐴 → (𝑧𝐴𝑧 ∈ suc 𝐴))
122, 10, 11e03 39284 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦𝐴   ▶   𝑧 ∈ suc 𝐴   )
1312in3 39151 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑧 ∈ suc 𝐴)   )
14 idn3 39157 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑦 = 𝐴   )
15 eleq2 2719 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
1615biimpcd 239 . . . . . . . . 9 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
176, 14, 16e23 39299 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧𝐴   )
182, 17, 11e03 39284 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ,   𝑦 = 𝐴   ▶   𝑧 ∈ suc 𝐴   )
1918in3 39151 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦 = 𝐴𝑧 ∈ suc 𝐴)   )
20 simpr 476 . . . . . . . 8 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
214, 20e2 39173 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑦 ∈ suc 𝐴   )
22 elsuci 5829 . . . . . . 7 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
2321, 22e2 39173 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   (𝑦𝐴𝑦 = 𝐴)   )
24 jao 533 . . . . . 6 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
2513, 19, 23, 24e222 39178 . . . . 5 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ suc 𝐴)   ▶   𝑧 ∈ suc 𝐴   )
2625in2 39147 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
2726gen12 39160 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)   )
28 biimpr 210 . . 3 ((Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) → Tr suc 𝐴))
291, 27, 28e01 39233 . 2 (   Tr 𝐴   ▶   Tr suc 𝐴   )
3029in1 39104 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  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-vd2 39111  df-vd3 39123
This theorem is referenced by: (None)
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