Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  trin Unicode version

Theorem trin 3864
 Description: The intersection of transitive classes is transitive. (Contributed by NM, 9-May-1994.)
Assertion
Ref Expression
trin

Proof of Theorem trin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3126 . . . . 5
2 trss 3863 . . . . . 6
3 trss 3863 . . . . . 6
42, 3im2anan9 530 . . . . 5
51, 4syl5bi 141 . . . 4
6 ssin 3159 . . . 4
75, 6syl6ib 150 . . 3
87ralrimiv 2391 . 2
9 dftr3 3858 . 2
108, 9sylibr 137 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wcel 1393  wral 2306   cin 2916   wss 2917   wtr 3854 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855 This theorem is referenced by:  ordin  4122
 Copyright terms: Public domain W3C validator