MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suctrOLD Structured version   Visualization version   GIF version

Theorem suctrOLD 5768
Description: Obsolete proof of suctr 5767 as of 24-Sep-2021. (Contributed by Alan Sare, 11-Apr-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
suctrOLD (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctrOLD
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 477 . . . . 5 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
2 vex 3189 . . . . . 6 𝑦 ∈ V
32elsuc 5753 . . . . 5 (𝑦 ∈ suc 𝐴 ↔ (𝑦𝐴𝑦 = 𝐴))
41, 3sylib 208 . . . 4 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
5 simpl 473 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
6 eleq2 2687 . . . . . . 7 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
75, 6syl5ibcom 235 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧𝐴))
8 elelsuc 5756 . . . . . 6 (𝑧𝐴𝑧 ∈ suc 𝐴)
97, 8syl6 35 . . . . 5 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
10 trel 4719 . . . . . . . . 9 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
1110expd 452 . . . . . . . 8 (Tr 𝐴 → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
1211adantrd 484 . . . . . . 7 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑧𝐴)))
1312, 8syl8 76 . . . . . 6 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑧 ∈ suc 𝐴)))
14 jao 534 . . . . . 6 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
1513, 14syl6 35 . . . . 5 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴))))
169, 15mpdi 45 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
174, 16mpdi 45 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1817alrimivv 1853 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
19 dftr2 4714 . 2 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2018, 19sylibr 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 1987  Tr wtr 4712  suc csuc 5684
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
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 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3560  df-in 3562  df-ss 3569  df-sn 4149  df-uni 4403  df-tr 4713  df-suc 5688
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator