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Theorem suctrALT2 38541
Description: Virtual deduction proof of suctr 5770. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 38540 using the tools command file translatewithout_overwritingminimize_excludingduplicates.cmd . (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
suctrALT2 (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrALT2
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sssucid 5764 . . . . 5 𝐴 ⊆ suc 𝐴
2 trel 4724 . . . . . . 7 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
32expd 452 . . . . . 6 (Tr 𝐴 → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
43adantrd 484 . . . . 5 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑧𝐴)))
5 ssel 3582 . . . . 5 (𝐴 ⊆ suc 𝐴 → (𝑧𝐴𝑧 ∈ suc 𝐴))
61, 4, 5ee03 38436 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑧 ∈ suc 𝐴)))
7 simpl 473 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
87a1i 11 . . . . . 6 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦))
9 eleq2 2693 . . . . . . 7 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
109biimpcd 239 . . . . . 6 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
118, 10syl6 35 . . . . 5 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧𝐴)))
121, 11, 5ee03 38436 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴)))
13 simpr 477 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
1413a1i 11 . . . . 5 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴))
15 elsuci 5753 . . . . 5 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
1614, 15syl6 35 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴)))
17 jao 534 . . . 4 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
186, 12, 16, 17ee222 38176 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1918alrimivv 1858 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
20 dftr2 4719 . 2 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2119, 20sylibr 224 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  wa 384  wal 1478   = wceq 1480  wcel 1992  wss 3560  Tr wtr 4717  suc csuc 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-un 3565  df-in 3567  df-ss 3574  df-sn 4154  df-uni 4408  df-tr 4718  df-suc 5691
This theorem is referenced by: (None)
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