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

Theorem ontrci 4344
 Description: An ordinal number is a transitive class. (Contributed by NM, 11-Jun-1994.)
Hypothesis
Ref Expression
on.1 𝐴 ∈ On
Assertion
Ref Expression
ontrci Tr 𝐴

Proof of Theorem ontrci
StepHypRef Expression
1 on.1 . . 3 𝐴 ∈ On
21onordi 4343 . 2 Ord 𝐴
3 ordtr 4295 . 2 (Ord 𝐴 → Tr 𝐴)
42, 3ax-mp 5 1 Tr 𝐴
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1480  Tr wtr 4021  Ord word 4279  Oncon0 4280 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285 This theorem is referenced by:  onunisuci  4349  exmidonfinlem  7042  bj-el2oss1o  12970  nnsf  13188
 Copyright terms: Public domain W3C validator