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Theorem suctr 4185
Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
Assertion
Ref Expression
suctr (Tr 𝐴 → Tr suc 𝐴)

Proof of Theorem suctr
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 107 . . . . 5 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑦 ∈ suc 𝐴)
2 vex 2577 . . . . . 6 𝑦 ∈ V
32elsuc 4170 . . . . 5 (𝑦 ∈ suc 𝐴 ↔ (𝑦𝐴𝑦 = 𝐴))
41, 3sylib 131 . . . 4 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑦 = 𝐴))
5 simpl 106 . . . . . . 7 ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧𝑦)
6 eleq2 2117 . . . . . . 7 (𝑦 = 𝐴 → (𝑧𝑦𝑧𝐴))
75, 6syl5ibcom 148 . . . . . 6 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧𝐴))
8 elelsuc 4173 . . . . . 6 (𝑧𝐴𝑧 ∈ suc 𝐴)
97, 8syl6 33 . . . . 5 ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦 = 𝐴𝑧 ∈ suc 𝐴))
10 trel 3888 . . . . . . . . 9 (Tr 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
1110expd 249 . . . . . . . 8 (Tr 𝐴 → (𝑧𝑦 → (𝑦𝐴𝑧𝐴)))
1211adantrd 268 . . . . . . 7 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑧𝐴)))
1312, 8syl8 69 . . . . . 6 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → (𝑦𝐴𝑧 ∈ suc 𝐴)))
14 jao 682 . . . . . 6 ((𝑦𝐴𝑧 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
1513, 14syl6 33 . . . . 5 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → ((𝑦 = 𝐴𝑧 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴))))
169, 15mpdi 42 . . . 4 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → ((𝑦𝐴𝑦 = 𝐴) → 𝑧 ∈ suc 𝐴)))
174, 16mpdi 42 . . 3 (Tr 𝐴 → ((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
1817alrimivv 1771 . 2 (Tr 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
19 dftr2 3883 . 2 (Tr suc 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴))
2018, 19sylibr 141 1 (Tr 𝐴 → Tr suc 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wo 639  wal 1257   = wceq 1259  wcel 1409  Tr wtr 3881  suc csuc 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-uni 3608  df-tr 3882  df-suc 4135
This theorem is referenced by:  ordsucim  4253  ordom  4356
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