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Theorem suctr 4124
Description: The successor of a transitive class is transitive. (Contributed by Alan Sare, 11-Apr-2009.)
Assertion
Ref Expression
suctr  Tr  Tr  suc

Proof of Theorem suctr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 103 . . . . 5  suc  suc
2 vex 2554 . . . . . 6 
_V
32elsuc 4109 . . . . 5  suc
41, 3sylib 127 . . . 4  suc
5 simpl 102 . . . . . . 7  suc
6 eleq2 2098 . . . . . . 7
75, 6syl5ibcom 144 . . . . . 6  suc
8 elelsuc 4112 . . . . . 6  suc
97, 8syl6 29 . . . . 5  suc  suc
10 trel 3852 . . . . . . . . 9  Tr
1110expd 245 . . . . . . . 8  Tr
1211adantrd 264 . . . . . . 7  Tr  suc
1312, 8syl8 65 . . . . . 6  Tr  suc  suc
14 jao 671 . . . . . 6  suc 
suc  suc
1513, 14syl6 29 . . . . 5  Tr  suc 
suc  suc
169, 15mpdi 38 . . . 4  Tr  suc  suc
174, 16mpdi 38 . . 3  Tr  suc  suc
1817alrimivv 1752 . 2  Tr  suc  suc
19 dftr2 3847 . 2  Tr 
suc  suc  suc
2018, 19sylibr 137 1  Tr  Tr  suc
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wo 628  wal 1240   wceq 1242   wcel 1390   Tr wtr 3845   suc csuc 4068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-uni 3572  df-tr 3846  df-suc 4074
This theorem is referenced by:  ordsucim  4192  ordom  4272
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