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Theorem ontrci 5831
 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 5830 . 2 Ord 𝐴
3 ordtr 5735 . 2 (Ord 𝐴 → Tr 𝐴)
42, 3ax-mp 5 1 Tr 𝐴
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1989  Tr wtr 4750  Ord word 5720  Oncon0 5721 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-v 3200  df-in 3579  df-ss 3586  df-uni 4435  df-tr 4751  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-ord 5724  df-on 5725 This theorem is referenced by:  onunisuci  5839  hfuni  32275
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