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Theorem suctrALT2 39571
Description: Virtual deduction proof of suctr 5969. The sucessor of a transitive class is transitive. This proof was generated automatically from the virtual deduction proof suctrALT2VD 39570 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 5963 . . . . 5 𝐴 ⊆ suc 𝐴
2 trel 4911 . . . . . . 7 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
32expd 451 . . . . . 6 (Tr 𝐴 → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
43adantrd 485 . . . . 5 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑧𝐴)))
5 ssel 3738 . . . . 5 (𝐴 ⊆ suc 𝐴 → (𝑧𝐴𝑧 ∈ suc 𝐴))
61, 4, 5ee03 39470 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑧 ∈ suc 𝐴)))
7 simpl 474 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
87a1i 11 . . . . . 6 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦))
9 eleq2 2828 . . . . . . 7 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
109biimpcd 239 . . . . . 6 (𝑧𝑦 → (𝑦 = 𝐴𝑧𝐴))
118, 10syl6 35 . . . . 5 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧𝐴)))
121, 11, 5ee03 39470 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴)))
13 simpr 479 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
1413a1i 11 . . . . 5 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴))
15 elsuci 5952 . . . . 5 (𝑦 ∈ suc 𝐴 → (𝑦𝐴𝑦 = 𝐴))
1614, 15syl6 35 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴)))
17 jao 535 . . . 4 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
186, 12, 16, 17ee222 39210 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1918alrimivv 2005 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
20 dftr2 4906 . 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 382  wa 383  wal 1630   = wceq 1632  wcel 2139  wss 3715  Tr wtr 4904  suc csuc 5886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-in 3722  df-ss 3729  df-sn 4322  df-uni 4589  df-tr 4905  df-suc 5890
This theorem is referenced by: (None)
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