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Theorem ontrci 6283
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 6282 . 2 Ord 𝐴
3 ordtr 6192 . 2 (Ord 𝐴 → Tr 𝐴)
42, 3ax-mp 5 1 Tr 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2115  Tr wtr 5158  Ord word 6177  Oncon0 6178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3138  df-v 3482  df-in 3926  df-ss 3936  df-uni 4825  df-tr 5159  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-ord 6181  df-on 6182
This theorem is referenced by:  onunisuci  6291  hfuni  33706
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