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Theorem ontrci 4389
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 4388 . 2 Ord 𝐴
3 ordtr 4340 . 2 (Ord 𝐴 → Tr 𝐴)
42, 3ax-mp 5 1 Tr 𝐴
Colors of variables: wff set class
Syntax hints:  wcel 2128  Tr wtr 4064  Ord word 4324  Oncon0 4325
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-uni 3775  df-tr 4065  df-iord 4328  df-on 4330
This theorem is referenced by:  onunisuci  4394  exmidonfinlem  7130  bj-el2oss1o  13419  nnsf  13648
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