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Theorem suctrOLD 5968
 Description: Obsolete proof of suctr 5967 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 479 . . . . 5 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
2 vex 3341 . . . . . 6 𝑦 ∈ V
32elsuc 5953 . . . . 5 (𝑦 ∈ suc 𝐴 ↔ (𝑦𝐴𝑦 = 𝐴))
41, 3sylib 208 . . . 4 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
5 simpl 474 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
6 eleq2 2826 . . . . . . 7 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
75, 6syl5ibcom 235 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧𝐴))
8 elelsuc 5956 . . . . . 6 (𝑧𝐴𝑧 ∈ suc 𝐴)
97, 8syl6 35 . . . . 5 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
10 trel 4909 . . . . . . . . 9 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
1110expd 451 . . . . . . . 8 (Tr 𝐴 → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
1211adantrd 485 . . . . . . 7 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑧𝐴)))
1312, 8syl8 76 . . . . . 6 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑧 ∈ suc 𝐴)))
14 jao 535 . . . . . 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 2003 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
19 dftr2 4904 . 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 382   ∧ wa 383  ∀wal 1628   = wceq 1630   ∈ wcel 2137  Tr wtr 4902  suc csuc 5884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-v 3340  df-un 3718  df-in 3720  df-ss 3727  df-sn 4320  df-uni 4587  df-tr 4903  df-suc 5888 This theorem is referenced by: (None)
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